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SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 21: Trigonometric graphs and
identities
Authored by:
Margaret Ferguson
Reviewed by:
Jillian Hornby
Previously authored by:
Eddie Mullan
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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SCHOLAR Study Guide Course Materials Topic 21: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
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1
Topic 1
Trigonometric equations, graphs
and identities
Contents
21.1 Sketching trigonometric graphs . . . . . . . . . . . . . . . . . . . . . . . . . .
3
21.2 Identifying trigonometric functions from graphs . . . . . . . . . . . . . . . . . .
21.3 Solving trigonometric equations . . . . . . . . . . . . . . . . . . . . . . . . . .
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32
21.4 Exact trigonometric values . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.5 Using trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . .
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46
21.6 Learning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.7 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Learning objectives
By the end of this topic, you should be able to:
•
identify the features of Sine, Cosine and Tangent graphs;
•
identify the amplitude and period of a trig function;
•
identify and sketch trig graphs with:
◦
multiple angles;
◦
vertical translations;
◦
horizontal translations;
•
solve trigonometric equations;
•
determine exact trigonometric values;
•
use trig identities to simplify trigonometric expressions.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
1.1
3
Sketching trigonometric graphs
Graphs of trigonometric functions (1)
This activity allows you to investigate the graphs of various types of trigonometric
functions. This will help you to be able to sketch the graphs of trigonometric functions
from their equation.
Using the equation y = a sin bx ◦ practice drawing graphs changing the values of a, b
and the type of trigonometric function (sin, cos and tan).
Here are some examples when the trigonometric function is sin. Notice how the values
of a and b change the graph of the function.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Key point
The Amplitude of a graph
The amplitude of a graph is half of the distance between the maximum and
minimum values of the graph.
For example: In y = a sin x and y = a cos x the amplitude is represented by a.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
5
Key point
The Period of a graph
The period of a graph is the distance along the x-axis over which the graph
completes one full wave pattern.
The period of y = sin x and y = cos x is 360 degrees.
Key point
y = tan x is different. . .
The graph is not a smooth curve.
The period of y = tan x is 180 ◦ .
Graphs of trigonometric functions practice
Q1: Sketch the graph of y = sin x.
..........................................
Q2: Sketch the graph of y = cos x.
..........................................
Q3: Sketch the graph of y = tan x.
..........................................
Examples
1.
Problem:
Sketch the graph of y = sin 3x◦ for 0 ≤ x ≤ 360.
Solution:
Step 1:
We want a ’sin’ wave which starts at the origin and rises to a maximum.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 2:
This maximum is 1 and the minimum is -1 since we want y = 1sin 3x ◦ .
The amplitude = 1.
Step 3:
We need 3 waves in 360 degrees for y = sin 3x ◦ .
Step 4:
The period = 360 ÷ 3 = 120◦ so each full wave covers a distance of 120 ◦ .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
2.
Problem:
Sketch the graph of y = 4 cos 2x◦ for 0 ≤ x ≤ 360.
Solution:
Step 1:
We want a ’cos’ wave which starts at the maximum and drops through zero to the
minimum.
Step 2:
This maximum is 4 and the minimum is -4 since we want y = 4 cos 2x ◦ .
The amplitude = 4
Step 3:
We need 2 waves in 360 degrees for y = 4 cos 2x ◦ .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 4:
The period = 360 ÷ 2 = 180◦ so each full wave covers a distance of 180 ◦ .
..........................................
3.
Problem:
Sketch the graph of y = − tan 2x◦ for 0 ≤ x ≤ 360.
Solution:
Step 1:
We need a ’tan’ graph which starts at the origin and has asymptotes at x = 90 ◦ and
x = 270◦ .
An asymptote is a line which the graph gets closer and closer to but never actually
touches.
The asymptotes will help us to make a better sketch.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
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Step 2:
The negative reflects the graph in the x-axis.
Step 3:
We require two copies of this tan graph in 360 ◦ for y = − tan 2x.
The period of y = tan x is 180 ◦ .
The period of y = − tan 2x is 180 ÷ 2 = 90 ◦ .
We now have asymptotes at x = 45, 135, 225 and 315.
..........................................
Key point
The graph of y = − sin x is the graph of y = sin x reflected in the x-axis.
Sketching trigonometric functions exercise 1
Q4: Sketch the graph of y = cos 4x◦ , for 0 ≤ x ≤ 360◦ . Remember the equation
takes the form y = a cos bx.
..........................................
Q5: Sketch the graph of y = 3 sin x ◦ , for 0 ≤ x ≤ 360◦ . Remember the equation
takes the form y = a sin bx.
..........................................
Q6: Sketch the graph of y = 5 tan 3x ◦ , for 0 ≤ x ≤ 360◦ . Remember the equation
takes the form y = a tan bx.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q7: Sketch the graph of y = 4 cos 2x◦ , for 0 ≤ x ≤ 360◦ . Remember the equation
takes the form y = a cos bx.
..........................................
Q8: Sketch the graph of y = 6 sin 4x ◦ , for 0 ≤ x ≤ 360◦ . Remember the equation
takes the form y = a sin bx.
..........................................
Graphs of trigonometric functions (2)
This activity allows you to investigate vertical translations of trigonometric functions.
Go online
Using the equation y = a sin bx ◦ + c practice drawing graphs changing the values of
a, b, c and the type of trigonometric function (sin, cos and tan).
Here are some examples when the trigonometric function is sin. Notice how the values
of a, b and c change the graph of the function.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Key point
The graph of y = sin x + c is the graph of y = sin x moved up vertically by c
units.
The graph of y = sin x − c is the graph of y = sin x moved down vertically by
c units.
The same movement will be applied for y = cos x and y = tan x.
..........................................
Examples
1.
Problem:
Sketch the graph of y = sin 2x◦ − 1, for 0 ≤ x ≤ 360◦ .
Solution:
Step 1:
Start by sketching the graph of y = sin 2x◦ .
The graph of y = sin 2x ◦ starts at the origin and rises to a maximum.
The graph of y = sin 2x ◦ has a maximum of 1 and a minimum of -1.
The amplitude = 1.
There are 2 waves in y = sin 2x ◦ .
The period = 360 ÷ 2 = 180◦ so each full wave covers a distance of 180 ◦ .
Step 2:
The graph of y = sin 2x ◦ − 1 is the graph of y = sin 2x◦ moved down vertically by 1
unit.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
It is much easier to draw a faint version of the graph of y = sin 2x first then make a
copy moved down vertically by 1 unit.
The maximum is 0 and the minimum is -2.
The amplitude = 1 as before.
The period = 360 ÷ 2 = 180◦ as before.
..........................................
2.
Problem:
Sketch the graph of y = 2 cos 3x◦ + 4, for 0 ≤ x ≤ 360◦
Solution:
Step 1:
Start by sketching the graph of y = 2 cos 3x◦ .
The graph of y = 2 cos 3x ◦ starts at 2 and decreases to a minimum.
The graph of y = 2 cos 3x ◦ has a maximum of 2 and a minimum of -2.
The amplitude = 2.
There are 3 waves in y = 2 cos 3x ◦ .
The period = 360 ÷ 3 = 120◦ so each full wave covers a distance of 120 ◦ .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 2:
The graph of y = 2 cos 3x◦ + 4 is the graph of y = 2 cos 3x◦ moved up vertically by
4 units.
The maximum is 6 and the minimum is 2.
The amplitude = 2 as before.
The period = 360 ÷ 3 = 120◦ as before.
..........................................
3.
Problem:
Sketch the graph of y = tan x ◦ + 3, for 0 ≤ x ≤ 360◦
Solution:
Step 1:
Start by sketching the graph of y = tan x ◦ .
The graph of y = tan x ◦ starts at the origin and increases infinitely.
The graph of y = tan x ◦ has no maximum or minimum.
There is no amplitude.
The tan graph is different from sin and cos. There are 2 waves in y = tan x ◦ .
The period = 360 ÷ 2 = 180◦ so each full wave covers a distance of 180 ◦ .
The graph has asymptotes where the it tends to positive and negative infinity. y =
tan x◦ has asymptotes at 90◦ and 270◦ .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
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Step 2:
The graph of y = tan x ◦ + 3 is the graph of y = tan x◦ moved up vertically by 3 units.
The period = 360 ÷ 2 = 180◦ as before.
The asymptotes are at 90 ◦ and 270◦ as before.
..........................................
Key point
An asymptote is a vertical line of the form x = k at which the function is
undefined. The graph gets closer and closer to the asymptote but never actually
touches it.
Sketching trigonometric functions exercise 2
Q9: Sketch the graph of y = cos 2x◦ + 2, for 0 ≤ x ≤ 360◦ . Remember the
equation takes the form y = a cos bx + c.
..........................................
Q10: Sketch the graph of y = 5 sin 4x ◦ + 1, for 0 ≤ x ≤ 360◦ . Remember the
equation takes the form y = a sin bx + c.
..........................................
Q11: Sketch the graph of y = tan 2x ◦ − 4, for 0 ≤ x ≤ 360◦ . Remember the
equation takes the form y = a tan bx + c.
..........................................
Q12: Sketch the graph of y = 3 sin 2x ◦ − 4, for 0 ≤ x ≤ 360◦ . Remember the
equation takes the form y = a sin bx + c.
..........................................
Q13: Sketch the graph of y = cos 4x ◦ − 2, for 0 ≤ x ≤ 360◦ . Remember the
equation takes the form y = a cos bx + c.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Graphs of trigonometric functions (3)
This activity allows you to investigate horizontal translations of trigonometric functions.
Go online
Using the equation y = a sin (x + b ◦ ) + c practice drawing graphs changing the values
of a, b, c and the type of trigonometric function (sin, cos and tan).
Here are some examples when the trigonometric function is sin. Notice how the values
of a, b and c change the graph of the function.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Examples
1.
Problem:
Sketch the graph of y = sin (x◦ − 45), for 0 ≤ x ≤ 360◦
Solution:
Step 1:
Start by sketching the graph of y = sin x◦ .
The graph of y = sin x ◦ starts at the origin and rises to a maximum.
The graph of y = sin x ◦ has a maximum of 1 and a minimum of -1.
The amplitude = 1.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
There is 1 wave in y = sin x◦ so the period = 360◦ .
Step 2:
The graph of y = sin (x ◦ − 45) is the graph of y = sin x◦ moved right by 45◦ .
The maximum, minimum, amplitude and period are unchanged.
It is much easier to draw a faint version of the graph of y = sin x ◦ first then make a
copy 45◦ to the right.
..........................................
2.
Problem:
Sketch the graph of y = 3 cos (x◦ + 30), for 0 ≤ x ≤ 360◦
Solution:
Step 1:
Start by sketching the graph of y = 3 cos x◦ .
The graph of y = 3 cos x ◦ starts at 3 and decreases to a minimum.
The graph of y = 3 cos x ◦ has a maximum of 3 and a minimum of -3.
The amplitude = 3.
There is 1 wave in y = 3 cos x◦ so the period = 360◦ .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 2:
The graph of y = 3 cos (x◦ + 30) is the graph of y = 3 cos x◦ moved left by 30◦ .
The maximum, minimum, amplitude and period are unchanged.
It is much easier to draw a faint version of the graph of y = 3 cos x ◦ first then make a
copy 30◦ to the left.
..........................................
3.
Problem:
Sketch the graph of y = 2 sin (x + 90◦ ) + 1, for 0 ≤ x ≤ 360◦
Solution:
Step 1:
Start by sketching the graph of y = 2 sin x◦ .
The graph of y = 2 sin x ◦ starts at the origin and increases to a maximum.
The graph of y = 2 sin x ◦ has a maximum of 2 and a minimum of -2.
The amplitude = 2.
There is 1 wave in y = 2 sin x◦ so the period = 360◦ .
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 2:
The graph of y = 2 sin x ◦ + 1 is the graph of y = 2 sin x◦ moved up vertically by 1
unit.
The graph of y = 2 sin x ◦ + 1 has a maximum of 3 and a minimum of -1.
The amplitude = 2 as before and the period is unchanged.
Step 3:
The graph of y = 2 sin (x + 90◦ ) + 1 is the graph of y = 2 sin x◦ + 1 moved left by
90◦ .
Note again that it is much easier to make faint copies of y = 2 sin x ◦ and y =
2 sin x◦ + 1 then make a copy 90 ◦ to the left.
Notice that the graph now looks like y = 2 cos x ◦ + 1.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
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Key point
The graph of y = sin(x + c) is the graph of y = sin x moved c degrees to the
left.
The graph of y = sin(x − c) is the graph of y = sin x moved c degrees to the
right.
The same movement will be applied for y = cos x and y = tan x.
Sketching trigonometric functions exercise 3
Q14: Sketch the graph of y = 6 sin (x ◦ − 15) − 2, for 0 ≤ x ≤ 360◦ . Remember
the equation takes the form y = a sin (x + b) + c.
Go online
..........................................
Q15: Sketch the graph of y = 2 cos (4x ◦ + 60) − 2, for 0 ≤ x ≤ 360◦ . Remember
the equation takes the form y = a cos (x + b) + c.
..........................................
Q16: Sketch the graph of y = sin (3x ◦ + 60) + 4, for 0 ≤ x ≤ 360◦ . Remember
the equation takes the form y = a sin (x + b) + c.
..........................................
Q17: Sketch the graph of y = tan (2x ◦ + 90) − 4, for 0 ≤ x ≤ 360◦ . Remember
the equation takes the form y = a tan (x + b) + c.
..........................................
Q18: Sketch the graph of y = 3 cos (2x ◦ + 90) + 3, for 0 ≤ x ≤ 360◦ . Remember
the equation takes the form y = a cos (x + b) + c.
..........................................
Q19: Sketch the graph of y = 3 tan (x ◦ + 45) − 1, for 0 ≤ x ≤ 360◦ . Remember
the equation takes the form y = a tan (x + b) + c.
..........................................
1.2
Identifying trigonometric functions from graphs
Before we identify the equations of trigonometric graphs use the activity below remind
yourself what the graphs of some trigonometric functions look like.
Graphs of trigonometric functions 1
Using the equation y = a sin bx practice drawing graphs changing the values of a and
b and the type of trigonometric function (sin, cos and tan).
Here are some examples, notice how the values of a and b change the graph of the
function.
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22
TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Example
Problem:
Here is a sketch of the graph of y = a sin bx◦ , 0 ≤ x ≤ 360.
What are the values of a and b?
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
23
Solution:
Step 1:
The graph goes between a maximum of 3 and a minimum of -3 so the amplitude = 3,
hence a = 3.
Step 2:
There are two complete waves in 360 ◦ so b = 2.
(Alternatively since one wave = 180 ◦ , b = 360 ÷ 180 = 2).
Hence the equation of the graph is y = 3 sin 2x ◦ .
..........................................
Identifying trigonometric functions exercise 1
Q20: Identify the trigonometric function, y = a sin bx◦ , 0 ≤ x ≤ 360, shown in the
graph.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q21: Identify the trigonometric function, y = a cos bx◦ , 0 ≤ x ≤ 360, shown in the
graph.
..........................................
Q22: Identify the trigonometric function, y = a sin bx◦ , 0 ≤ x ≤ 360, shown in the
graph.
..........................................
Q23: Identify the trigonometric function, y = a cos bx◦ , 0 ≤ x ≤ 360, shown in the
graph.
..........................................
Before we identify the equations of other trigonometric graphs use the activity below to
remind yourself what the graphs of trigonometric functions with vertical translations look
like.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
25
Graphs of trigonometric functions 2
Using the equation y = a sin bx + c practice drawing graphs changing the values of a,
b and c and the type of trigonometric function (sin, cos and tan).
Here are some examples, notice how the values of a, b and c change the graph of the
function.
..........................................
Example
Problem:
Here is a sketch of the graph of y = a sin bx◦ + c, 0 ≤ x ≤ 360.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
What are the values of a, b and c?
Solution:
Step 1:
The graph goes between a maximum of 6 and a minimum of -4 so the amplitude =
6 - (-4) / = 5, hence a = 5.
2
Step 2:
There are two complete waves in 360 ◦ so b = 2.
(Alternatively since one wave = 180 ◦ , b = 360 ÷ 180 = 2).
Step 3:
So far we think the equation is y = 5 sin 2x ◦ .
The graph of y = 5 sin 2x ◦ has a maximum of 5 and a minimum of -5 but the graph in
the problem has a maximum of 6 and a minimum of -4.
The graph has been moved up by 1 unit so c = 1.
Hence the equation of the graph is y = 5sin 2x ◦ + 1
..........................................
Identifying trigonometric functions exercise 2
Go online
Q24: Identify the trigonometric function, y = a sin bx◦ + c, 0 ≤ x ≤ 360, shown in
the graph.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q25: Identify the trigonometric function, y = a cos bx◦ + c, 0 ≤ x ≤ 360, shown in
the graph.
..........................................
Q26: Identify the trigonometric function, y = a cos bx◦ + c, 0 ≤ x ≤ 360, shown in
the graph.
..........................................
Q27: Identify the trigonometric function, y = a sin bx◦ + c, 0 ≤ x ≤ 360, shown in
the graph.
..........................................
Before we identify the equations of other trigonometric graphs use the activity below to
remind yourself what the graphs of trigonometric functions with horizontal translations
look like.
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Graphs of trigonometric functions 3
Go online
Using the equation y = a sin(x + b) + c practice drawing graphs changing the values
of a, b and c and the type of trigonometric function (sin, cos and tan).
Here are some examples, notice how the values of a, b and c change the graph of the
function.
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Examples
1.
Problem:
Here is a sketch of the graph of y = a sin (x + b◦ ) + c, 0 ≤ x ≤ 360.
What are the values of a, b and c?
Solution:
Step 1:
The graph goes between a maximum of 3 and a minimum of 1 so the amplitude =
3 - 1 / = 1, hence a = 1.
2
Step 2:
There is one complete wave in 360 ◦ but the graph of y = sin x usually starts at the
origin and this one does not.
It starts at -30◦ .
The graph has been moved left by 30 ◦ , hence b = 30.
Step 3:
So far we think the equation is y = 1 sin (x + 30 ◦ ).
The graph of y = 1 sin (x + 30◦ ) has a maximum of 1 and a minimum of -1 but the
graph in the problem has a maximum of 3 and a minimum of 1.
The graph has been moved up by 2 units so c = 2.
Hence the equation of the graph is y = sin(x + 30 ◦ ) + 2
..........................................
2.
Problem:
Here is a sketch of the graph of y = a cos (x − b◦ ) + c, 0 ≤ x ≤ 360.
What are the values of a, b and c?
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Solution:
Step 1:
The graph goes between a maximum of 1 and a minimum of -3 so the amplitude =
1 - (-3) / = 2, hence a = 2.
2
Step 2:
There is one complete wave in 360 ◦ but the graph of y = cos x usually starts at its
maximum and this one does not.
You should be able to see where the graph is at its maximum i.e. 60 ◦ .
The graph has been moved right by 60 ◦ , hence b = 60.
Step 3:
So far we think the equation is y = 2 cos (x − 60 ◦ ).
The graph of y = 2 cos (x − 60◦ ) has a maximum of 2 and a minimum of -2 but the
graph in the problem has a maximum of 1 and a minimum of -3.
The graph has been moved down by 1 unit so c = − 1.
Hence the equation of the graph is y = 2cos(x − 60 ◦ ) − 1.
..........................................
Identifying trigonometric functions exercise 3
Go online
Q28: Identify the trigonometric function, y = a sin (x◦ + b) + c, 0 ≤ x ≤ 360, shown
in the graph.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q29: Identify the trigonometric function, y = a cos (x◦ + b) + c, 0 ≤ x ≤ 360, shown
in the graph.
..........................................
Q30: Identify the trigonometric function, y = a sin (x◦ + b) + c, 0 ≤ x ≤ 360, shown
in the graph.
..........................................
Q31: Identify the trigonometric function, y = a cos (x◦ + b) + c, 0 ≤ x ≤ 360, shown
in the graph.
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
1.3
Solving trigonometric equations
Solving trigonometric equations
Go online
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Key point
To summarise, when solving a trig equation there are two solutions for 0 ≤ x ≤
360◦ .
Using the inverse trig functions on your calculator, you will get the acute angle a
then:
•
for a sin equation the second solution is 180 − a;
•
for a tan equation the second solution is 180 + a.
•
for a cos equation the second solution is 360 − a;
This is often memorised by the aid of a diagram.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Top tip
There are many ways to remember the quadrant diagram e.g.
•
Going anti-clockwise round the quadrants this give ASTC . . . All Sinners
Take Care.
•
Going anti-clockwise round the quadrants gives CAST.
Key point
You should draw a little quadrant chart every time you solve a trig equation to
identify where you will find solutions.
Examples
1.
Problem:
Solve the equation 3 sin x ◦ = 2, 0 ≤ x ≤ 360◦
Solution:
Step 1:
Rearrange the equation 3 sin x ◦ = 2 ⇒ sin x◦ =
2
3
Step 2:
Draw a quadrant chart and tick the quadrants where sin has solutions, i.e. all and sin.
Step 3:
-1
Use your
calculator◦ to find the first solution [using the sin button] ⇒
−1 2
= 41 · 8
sin
3
x
=
Step 4:
Find the second solution [for sin use 180 − a] ⇒ x = 180 − 41 · 8 = 138 · 2◦
..........................................
2.
Problem:
Solve the equation 5 cos x ◦ = 4, 0 ≤ x ≤ 360◦ .
Solution:
Step 1:
Rearrange the equation 5 cos x ◦ = 4 ⇒ cos x◦ =
4
/ 5.
Step 2:
Draw a quadrant chart and tick the quadrants where cos has solutions, i.e. all and cos.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 3:
-1
Use your
4 calculator◦ to find the first solution [using the cos button] ⇒
−1
= 36 · 9
cos
5
35
x
=
Step 4:
Find the second solution [for cos use 360 − a] ⇒ x = 360 − 36 · 9 = 323 · 1◦
..........................................
3.
Problem:
Solve the equation 3 tan x ◦ = 5, 0 ≤ x ≤ 360◦
Solution:
Step 1:
Rearrange the equation 3 tan x ◦ = 5 ⇒ tan x◦ =
5
3
Step 2:
Draw a quadrant chart and tick the quadrants where tan has solutions, i.e. all and tan.
Step 3:
-1
Use your
5 calculator◦ to find the first solution [using the tan button] ⇒
−1
= 59 · 0
tan
3
x
=
Step 4:
Find the second solution [tan so use 180 + a] ⇒ ⇒ x = 180 + 59 · 0 = 239 · 0 ◦
..........................................
Solving trigonometric equations practice (1)
Q32:
Solve the equation 4 sin x ◦ = 1, 0 ≤ x ≤ 360◦
..........................................
Q33:
Solve the equation 5 cos x ◦ = 3, 0 ≤ x ≤ 360◦
..........................................
Q34:
Solve the equation 7 tan x ◦ = 2, 0 ≤ x ≤ 360◦
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
So far we have looked for solutions where sin x, cos x and tan x equal a positive
number.
If sin x, cos x and tan x equal a negative number we must look for solutions in the
opposite quadrants.
If we take the graph of y = sin x you will see that the graph is positive between 0 ◦ and
180◦ and that corresponds to the All and Sin quadrants. The graph is negative between
180◦ and 360◦ and that corresponds to the T an and Cos quadrants.
You know that cos x is positive in the All and Cos quadrants so it follows that cos x is
negative in the Sin and T an quadrants.
You know that tan x is positive in the All and T an quadrants so it follows that tan x is
negative in the Sin and Cos quadrants.
Example
Problem:
Solve the equation 5 sin x ◦ + 4 = 0, 0 ≤ x ≤ 360◦
Solution:
Step 1:
Rearrange the equation 5 sin x ◦ + 4 = 0 ⇒ sin x◦ = − 45 .
Step 2:
Draw a quadrant chart and tick the quadrants where sin has negative solutions, i.e. tan
and cos.
Step 3:
Use your calculator
to find the solution for sin x =
sin-1 = 45 = 53 · 1◦
4
5
[using the sin-1 button] ⇒ x =
Step 4:
Find the first solution [it’s the one in the tan quadrant so use 180 + a] ⇒
180 + 53 · 1 = 233 · 1◦
x
=
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Step 5:
Find the second solution [it’s the one in the cos quadrant so use 360 − a] ⇒
360 − 53 · 1 = 306 · 9◦
37
x
=
..........................................
Key point
It is extremely important to underline all your solutions when solving trig
equations.
Key point
Always find the acute angle in the all quadrant first. This means never putting a
negative value into your calculator to find the inverse trig value.
Solving trigonometric equations practice (2)
Q35:
Solve the equation 2 sin x ◦ = − 1, 0 ≤ x ≤ 360◦
Go online
..........................................
Solving trigonometric equations exercise
Q36: Solve the equation 6 sin x = 1, 0 ≤ x ≤ 360 ◦
..........................................
Q37: Solve the equation 6 cos x = − 4, 0 ≤ x ≤ 360 ◦
..........................................
Q38: Solve the equation tan x = − 2, 0 ≤ x ≤ 360 ◦
..........................................
Q39: Solve the equation 8 sin x + 1 = 4, 0 ≤ x ≤ 360 ◦
..........................................
Q40: Solve the equation 9 cos x − 2 = 0, 0 ≤ x ≤ 360 ◦
..........................................
Q41: Solve the equation 7 tan x + 2 = 10, 0 ≤ x ≤ 360 ◦
..........................................
Q42: Solve the equation 5 sin x = − 4, 0 ≤ x ≤ 360 ◦
..........................................
Q43: Solve the equation 3 cos x − 2 = − 4, 0 ≤ x ≤ 360 ◦
..........................................
Q44: Solve the equation 2 tan x + 7 = 0, 0 ≤ x ≤ 360 ◦
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
1.4
Exact trigonometric values
Knowing the exact values of the trig ratios sin, cos and tan for certain angles can allow
you to solve trig equations without a calculator.
When x = 0 the values of sin x, cos x and tan x are easily seen from their graphs.
Similarly when x = 90, 180, 270 and 360 the values of sin x, cos x and tan x can be
identified from their graphs.
The values are:
sin x
cos x
tan x
x = 0
0
1
0
x = 90◦
1
0
undefined
180◦
0
-1
0
x = 270◦
-1
0
undefined
360◦
0
1
0
x =
x =
By constructing two triangles it will be possible to determine the exact values of some
other common angles.
Construction of 30◦ and 60◦ angles
Sketch an equilateral triangle with side lengths of 2 units.
Go online
Draw in the perpendicular bisector from the apex to the base, focus on one of the
right-angled triangles now formed.
sides of the equilateral triangle are 2 units so the
√ base of the right-angled triangle is 1
unit (base halved by bisector) and the height is 3 units (by Pythagoras).
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
height =
=
=
√
√
39
22 − 12
4 − 1
3
Read off the exact values for sin, cos and tan for the angles 30 ◦ and 60◦ using the
SOH-CAH-TOA ratios.
..........................................
Construction of 45◦ angles
Sketch a square with side length 1 unit.
Go online
Draw in one of the diagonals, focus on one of the triangles now formed.
The sides of the square are 1 unit√so the base and the height of the right-angled triangle
are 1 unit and the hypotenuse is 2 units (by Pythagoras).
12 + 12
hypotenuse =
√
=
1 + 1
√
2
=
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Read off the exact values for sin, cos and tan for the angles 30 ◦ and 60◦ using the
SOH-CAH-TOA ratios.
..........................................
The table below shows all the exact values between 0 ◦ and 90◦ .
angle
0◦
sin x◦
0
cos x◦
tan x◦
1
0
30◦
45◦
60◦
90◦
1
2
√
3
2
√1
3
√1
2
3
2
1
1
2
0
√1
2
1
√
√
3
undefined
Key point
You will need to learn this table or remember how to construct the two special
triangles.
These exact values for trig. ratios can be extended to angles in all four quadrants.
Recall the quadrant diagram.
This can be remembered easily as All Students Talk Constantly and depicts the
quadrants in which the various ratios are positive.
The first quadrant has all
ratios (A) positive.
The second quadrant has
sine (S) positive.
The third quadrant has
tangent (T) positive.
The fourth quadrant has
cosine (C) positive.
To find an exact value for an angle greater than 90 ◦ , the angle is converted to the
associated acute angle which lies between 0 and 90 in the All quadrant. You should
recognise the diagram below from solving trig equations.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
The following examples show how to find the associated acute angle which will be
needed to determine the exact value.
Examples
1.
Problem:
Find the associated acute angle for 120 ◦ .
Solution:
120◦ is in the sin quadrant.
The related angle is 60 ◦ because 180 − 60 = 120◦ .
So the associated acute angle a = 60 ◦ .
..........................................
2.
Problem:
Find the associated acute angle for 300 ◦ .
Solution:
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
300◦ is in the cos quadrant.
The related angle is 60 ◦ because 360 − 60 = 300◦ .
So the associated acute angle a = 60 ◦ .
..........................................
3.
Problem:
Find the associated acute angle for 210 ◦ .
Solution:
210◦ is in the tan quadrant.
The related angle is 30 ◦ because 180 + 30 = 210 ◦ .
So the associated acute angle a = 30 ◦ .
..........................................
Exact trigonometric values practice
Q45: Find the associated acute angle for 315 ◦ .
Go online
..........................................
Q46: Find the associated acute angle for 150 ◦ .
..........................................
Q47: Find the associated acute angle for 240 ◦ .
..........................................
Exact values
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Example
Problem:
Find the exact value of cos 300◦ .
Solution:
300◦ is in quadrant 4.
Only cos is positive in the C quadrant so cos 300◦ is positive.
The associated acute angle is found by 360 − a = 300 ◦ so a = 60◦
cos 300◦ = cos 60◦
1
=
2
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Example
Problem:
Find the exact value of sin 330◦ .
Solution:
330◦ is in quadrant 4.
Only cos is positive in the C quadrant so sin 330◦ is negative.
The associated acute angle is found by 360 − a = 330 ◦ so a = 30◦
sin 330◦ = − sin 30◦
1
= −
2
..........................................
Exact trigonometric values practice
Go online
Q48: Find the exact value of sin 150◦ .
Is sin 150◦ positive or negative?
a) Positive
b) Negative
..........................................
Q49: What is the associated acute angle?
..........................................
Q50: What is the exact value of sin 150◦ ?
a)
b)
c)
√1
2
1
2
− 12
..........................................
Q51: Find the exact value of tan 150 ◦ .
Is tan 150◦ positive or negative?
a) Positive
b) Negative
..........................................
Q52: What is the associated acute angle?
..........................................
Q53: What is the exact value of tan 150◦ ?
a)
√1
√3
b)
3
√
c) − 3
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
45
Exact trigonometric values exercise
Q54: Construct the two special triangles or recall the exact values to answer the
following:
a) sin 60
b) cos 30
c) tan 45
d) sin 270
e) cos180
f) tan 360
..........................................
Q55: Construct the two special triangles or recall the exact values to answer the
following:
a) sin 150
b) cos 330
c) tan 210
..........................................
Q56: Construct the two special triangles or recall the exact values to answer the
following:
a) sin 210
b) sin 300
c) sin 315
..........................................
Q57: Construct the two special triangles or recall the exact values to answer the
following:
a) sin 120
b) cos 300
c) tan 240
..........................................
Q58: Construct the two special triangles or recall the exact values to answer the
following:
a) cos 150
b) cos 315
c) cos 135
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q59: Construct the two special triangles or recall the exact values to answer the
following:
a) sin 135
b) cos 315
c) tan 225
..........................................
Q60: Construct the two special triangles or recall the exact values to answer the
following:
a) tan 330
b) tan120
c) tan 315
..........................................
1.5
Using trigonometric identities
There are two trigonometric formulae to learn.
From SOHCAHTOA we know that: sin x =
O/
sin x
H
=
A/
cos x
H
H
O
×
=
H
A
O
=
A
= tan x
2
O
H
and cos x =
A
H
2
O 2
A
+
note: sin2 x means (sin x)2
H
H
O2
A2
+
H2
H2
2
O + A2
H2
2
H
by the Theorem of Pythagoras
H2
1
2
sin x + cos x =
=
=
=
=
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Key point
You will need to learn these trigonometric identities.
1.
sin x
cos x
= tan x
2. sin2 x + cos2 x = 1
Note that rearranging gives:
⇒ sin2 x = 1 − cos2 x
⇒ cos2 x = 1 − sin2 x
Examples
1.
Problem:
Simplify 4 sin2 x + 4cos2 x
Solution:
4 sin2 x + 4 cos2 x = 4 sin2 x + cos2 x
we know that sin2 x + cos2 x = 1
= 4 × 1
= 4
..........................................
2.
Problem:
Solve 4 sin x − 5 cos x = 0
Solution:
Divide each term by cos x.
4 sin x
5 cos x
= cos0 x
cos x − cos x
sin x
cos x
now cos
x = tan x, cos x
= 1 and
0
cos x
= 0 so,
4 tan x − 5 = 0
4 tan x = 5
5
tan x =
4
5
= 51 · 3◦
x = tan
4
x = 180 + 51 · 3 = 231 · 3◦
..........................................
−1
3.
Problem:
Prove that sin A sin2 x + sin A cos2 x = sinA
Solution:
The aim is to start with the left hand side of the equation and end up with the right hand
side of the equation.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
LHS = sin A sin2 x + sin A cos2 x
= sin A sin2 x + cos2 x
= sin A × 1
= sin A
= RHS
This is called a mathematical proof.
..........................................
Using trigonometric identities practice
Q61: Prove that tan a sin a =
Go online
1 − cos2 a
cos a
..........................................
Using trigonometric identities exercise
Q62: Using sin2 x + cos2 x = 1 and tan x =
Go online
sin x
cos x .
a) What is sin2 x equal to?
b) What is cos2 x equal to?
c) What is sin x equal to?
d) What is cos x equal to?
..........................................
Q63: Simplify:
a) 2 cos2 x + 2 sin2 x
b)
3 sin A
3 cos A
..........................................
Q64: Simplify:
a) (1 + cos x) (1 − cos x)
b) 2 − 2 sin2 A
..........................................
Q65: Prove that:
a) (sin X + cos X)2 = 1 + 2 sin X cos X
b) (sin x − cos x)2 + (sin x + cos x)2 = 2
c) cos2 A − sin2 A = 2 cos2 A − 1
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
1.6
Learning Points
Sketching and identifying trigonometric graphs
•
The graph of y = sin x looks like this:
•
The graph of y = cos x looks like this:
•
The graph of y = tan x looks like this:
•
The graph of y =
•
•
a sin bx has:
◦
a maximum of a;
◦
a minimum of −a;
◦
b complete waves in 360 ◦ .
The graph of y =
a cos bx +
◦
a maximum of a + c;
◦
a minimum of −a + c;
◦
b complete waves in 360 ◦ .
The graph of y =
a sin(x +
◦
a maximum of a + c;
◦
a minimum of −a + c;
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c has:
b) +
c has:
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
◦
been moved left or right horizontally by b ◦ .
•
The amplitude of a graph is half of the distance between the maximum and
minimum values of the graph.
•
The period of a graph is the distance along the x-axis over which the graph
completes one full wave pattern.
•
When identifying a trig graph:
◦
determine whether it is sin, cos or tan from its shape;
◦
identify the maximum and minimum values;
◦
determine the amplitude;
◦
identify the number of complete waves in 360 ◦ ;
◦
determine any vertical translation up or down;
◦
determine any horizontal translation left or right.
Solving trigonometric equations
•
Rearrange the equation to sin x ◦ = or tan x◦ = or cos x◦ =
•
Draw a quadrant chart and tick the quadrants where the function has solutions.
•
Use your calculator to find the acute angle
•
•
◦
Use the sin-1 or tan-1 or cos-1 button.
◦
Never enter a negative to find the inverse trig value.
Find the other solution(s):
◦
for the sin quadrant use 180 − a;
◦
for the tan quadrant use 180 + a;
◦
for the cos quadrant use 360 − a;
Underline your solutions.
Exact trigonometric values
•
Learn the table of exact values between 0 ◦ and 90◦ or remember how to construct
the two special triangles.
angle
•
0◦
sin x◦
0
cos x◦
1
tan x◦
0
30◦
45◦
60◦
90◦
1
2
√
3
2
√1
2
3
2
1
1
2
0
√1
3
√1
2
1
√
√
3
undefined
Or learn how to draw the 2 special triangles.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
•
•
Trigonometric identities
•
sin2 x + cos2 x = 1
•
tan x =
sin x
cos x
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
1.7
End of topic test
End of topic 21 test
Sketching Trigonometric Graphs
Go online
Q66: Sketch the graph of y = cos 2x ◦ , for 0 ≤ x ≤ 360◦
..........................................
Q67: Sketch the graph of y = sin 3x ◦ , for 0 ≤ x ≤ 360◦
..........................................
Q68: Sketch the graph of y = tan 3x ◦ , for 0 ≤ x ≤ 360◦
..........................................
Q69: Sketch the graph of y = 4 sin 2x ◦ , for 0 ≤ x ≤ 360◦
..........................................
Q70: Sketch the graph of y = 4 tan 3x ◦ , for 0 ≤ x ≤ 360◦
..........................................
Q71: Sketch the graph of y = 3 cos 4x ◦ , for 0 ≤ x ≤ 360◦
..........................................
Q72: Sketch the graph of y = 3 sin 2x ◦ + 4, for 0 ≤ x ≤ 360◦
..........................................
Q73: Sketch the graph of y = 1 cos 4x ◦ − 3, for 0 ≤ x ≤ 360◦
..........................................
Q74: Sketch the graph of y = 1 tan 3x ◦ − 4, for 0 ≤ x ≤ 360◦
..........................................
Q75: Sketch the graph of y = 3 sin (x ◦ − 30), for 0 ≤ x ≤ 360◦
..........................................
Q76: Sketch the graph of y = − 3 cos (x ◦ + 90), for 0 ≤ x ≤ 360◦
..........................................
Q77: Sketch the graph of y = − tan (x ◦ − 60), for 0 ≤ x ≤ 360◦
..........................................
Q78: Sketch the graph of y = 2 tan (x ◦ + 90) + 2, for 0 ≤ x ≤ 360◦
..........................................
Q79: Sketch the graph of y = 3 cos (x ◦ + 15) − 1, for 0 ≤ x ≤ 360◦
..........................................
Q80: Sketch the graph of y = sin (x ◦ − 90) + 2, for 0 ≤ x ≤ 360◦
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Identifying Trigonometric Graphs
Q81:
a) Identify the trigonometric function, y = sin ax◦ , 0 ≤ x ≤ 360, shown in the graph
above.
b) What is the period?
..........................................
Q82:
a) Identify the trigonometric function, y = cos ax◦ , 0 ≤ x ≤ 360, shown in the
graph above.
b) What is the period?
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q83:
a) Identify the trigonometric function, y = a cos bx◦ , 0 ≤ x ≤ 360, shown in the
graph above.
b) What is the amplitude?
..........................................
Q84:
a) Identify the trigonometric function, y = a sin bx◦ , 0 ≤ x ≤ 360, shown in the
graph above.
b) What is the amplitude?
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q85:
a) Identify the trigonometric function, y = a cos bx◦ + c, 0 ≤ x ≤ 360, shown in
the graph above.
b) What is the amplitude?
..........................................
Q86:
a) Identify the trigonometric function, y = a sin bx◦ + c, 0 ≤ x ≤ 360, shown in the
graph above.
b) What is the amplitude?
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q87:
a) Identify the trigonometric function, y = a tan bx◦ + c, 0 ≤ x ≤ 360, shown in the
graph above.
b) What is the amplitude?
..........................................
Q88:
Identify the trigonometric function, y = a cos (x◦ + b), 0 ≤ x ≤ 360, shown in the
graph.
..........................................
Q89:
Identify the trigonometric function, y = a tan (x◦ + b), 0 ≤ x ≤ 360, shown in the
graph.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Q90:
Identify the trigonometric function, y = a sin (x◦ + b), 0 ≤ x ≤ 360, shown in the
graph.
..........................................
Q91:
Identify the trigonometric function, y = a tan (x◦ + b) + c, 0 ≤ x ≤ 360, shown in
the graph.
..........................................
Q92:
Identify the trigonometric function, y = a sin (x◦ + b) + c, 0 ≤ x ≤ 360, shown in
the graph.
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
..........................................
Q93:
Identify the trigonometric function, y = a cos (x◦ + b) + c, 0 ≤ x ≤ 360, shown in
the graph.
..........................................
Solving Trigonometric Graphs
Q94: Solve sin x = 0 · 18
..........................................
Q95: Solve 5 cos x = 2
..........................................
Q96: Solve 6 tan x − 2 = 0
..........................................
Q97: Solve 4 sin x = − 1
..........................................
Q98: Solve 7 cos x + 5 = 0
..........................................
Exact Trigonometric Graphs
Give the exact values the following.
Q99: sin 30◦
..........................................
Q100: cos 45◦
..........................................
Q101: tan 60◦
..........................................
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TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES
Q102: sin 135◦
..........................................
Q103: tan 330◦
..........................................
Q104: cos 240◦
..........................................
Trigonometric Identities
Q105: Simplify 2cos2 Y + 2sin2 Y
..........................................
Q106: Simplify
sin3 A
cos3 A
..........................................
Q107: Solve 4 cos x + 5 sin x = 0
..........................................
Q108: Prove that (3cos x + 2sin x)2 + (2cos x − 3sin x)2 = 13
..........................................
..........................................
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GLOSSARY
Glossary
perpendicular
a line is perpendicular to another if they meet at right angles (90 degrees)
perpendicular bisector
a perpendicular bisector passes through the midpoint of the line it is perpendicular
to
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ANSWERS: TOPIC 21
Answers to questions and activities
21 Trigonometric equations, graphs and identities
Graphs of trigonometric functions practice (page 5)
Q1:
Step 1:
Start by plotting the important points from the sin curve on your graph. It starts at the
origin and increases to the maximum value.
Step 2:
Sketch a smooth curve and label it.
Q2:
Step 1:
Start by plotting the important points from the cos curve on your graph. It starts at the
maximum value and decreases.
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ANSWERS: TOPIC 21
Step 2:
Sketch a smooth curve and label it.
Q3:
Step 1:
Start by plotting the important points on the tan curve and the dotted lines where the
graph is undefined. It starts at the origin and increases but remember the tan curve is
not continuous.
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ANSWERS: TOPIC 21
Step 2:
Sketch smooth curves and a label.
Sketching trigonometric functions exercise 1 (page 9)
Q4:
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Q5:
Q6:
Q7:
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ANSWERS: TOPIC 21
Q8:
Sketching trigonometric functions exercise 2 (page 15)
Q9:
Q10:
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ANSWERS: TOPIC 21
Q11:
Q12:
Q13:
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ANSWERS: TOPIC 21
Sketching trigonometric functions exercise 3 (page 21)
Q14:
Q15:
Q16:
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Q17:
Q18:
Q19:
Identifying trigonometric functions exercise 1 (page 23)
Q20: y = -4 sin 3·5x
Q21: y = cos 2·5x
Q22: y = 5 sin 0·5x
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ANSWERS: TOPIC 21
Q23: y = -6 cos 4x
Identifying trigonometric functions exercise 2 (page 26)
Q24: y = 3 sin 2·5x - 3
Q25: y = 2 cos 2·5x - 4
Q26: y = -2 cos 0·5x - 2
Q27: y = -2 sin 3·5 + 4
Identifying trigonometric functions exercise 3 (page 30)
Q28: y = -4 sin(x + 45) -1
Q29: y = -cos(x + 60) + 2
Q30: y = -2 sin(x + 45) + 3
Q31: y = 4 cos(x - 30) + 1
Solving trigonometric equations practice (1) (page 35)
Q32:
4 sin x◦ = 1
1
sin x◦ =
4
1
= 14 · 5◦
x = sin
4
x◦ = 180 − 14 · 5 = 165 · 5◦
◦
−1
Q33:
5 cos x◦ = 3
3
cos x◦ =
5
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ANSWERS: TOPIC 21
3
= 53 · 1◦
5
= 360 − 53 · 1 = 306 · 9◦
x◦ = cos−1
x◦
Q34:
7 tan x◦ = 2
2
tan x◦ =
7
x
◦
= tan
−1
2
= 15 · 9◦
7
x◦ = 180 + 15 · 9 = 195 · 9◦
Solving trigonometric equations practice (2) (page 37)
Q35:
2 sin x◦ = −1
1
sin x◦ = −
2
When sin x = 12 , x = sin−1 12
=
solutions in the tan and cos quadrants so,
30◦ but this is not a solution. We need
x = 180 + 30 = 210◦
x = 360 − 30 = 330◦
Solving trigonometric equations exercise (page 37)
Q36: x = 9 · 6◦ and x = 170 · 4◦
Q37: x = 131 · 8◦ and x = 228 · 2◦
Q38: x = 296 · 6◦ and x = 116 · 6◦
Q39:
Steps:
•
Re-arrange the equation.
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ANSWERS: TOPIC 21
•
sin x = ? 3 /8
•
Use this answer to find your solution remembering to press the sin -1 button.
Answer: x = 22 · 0◦ and x = 158 · 0◦
Q40:
Steps:
•
Re-arrange the equation.
•
cos x =? 2 /9
•
Use this answer to find your solution remembering to press the cos -1 button.
Answer: x = 77 · 2◦ and x = 282 · 8◦
Q41:
Steps:
•
Re-arrange the equation.
•
tan x = ? 8 /7
•
Use this answer to find your solution remembering to press the tan -1 button.
Answer: x = 48 · 8◦ and x = 228 · 8◦
Q42: x = 306 · 8◦ and x = 233 · 1◦
Steps:
•
Re-arrange the equation.
•
sin x = ? −4 /5
•
sin-1 (4 /5 ) = ? 53·1
•
Remember to identify the quadrants where sin x is negative and use this answer to
find your solutions.
Answer:
Q43:
Steps:
•
Re-arrange the equation.
•
cos x = ? −2 /3
•
cos-1 (2 /3 ) = ? 41·8
•
Remember to identify the quadrants where cos x is negative and use this answer
to find your solutions.
Answer: x = 318 · 2◦ and x = 41 · 8◦
Q44:
Steps:
•
Re-arrange the equation.
•
tan x = ? −7 /2
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ANSWERS: TOPIC 21
•
tan-1 (7 /2 ) = ? 74·1
•
Remember to identify the quadrants where tan x is negative and use this answer
to find your solutions.
Answer: x = 285 · 9◦ and x = 105 · 9◦
Exact trigonometric values practice (page 42)
Q45: 45 ◦
Q46: 30 ◦
Q47: 60 ◦
Exact trigonometric values practice (page 44)
Q48: a) Positive
Q49: 30 ◦
Q50: b)
1
2
Q51: b) Negative
Q52: 30 ◦
√
Q53: c) − 3
Exact trigonometric values exercise (page 45)
Q54:
√
a)
b)
3
2
√
3
2
c) 1
d) -1
e) -1
f) 0
Q55:
a) 1/2
√
b)
c)
3
2
√1
3
Q56:
a) -1/2
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ANSWERS: TOPIC 21
√
3
2
− √12
b) −
c)
Q57:
√
3
2
a)
b) 1/2
√
c) 3
Q58:
√
3
2
− √12
− √12
a) −
b)
c)
Q59:
a)
b)
√1
2
1
√
2
c) 1
Q60:
a) − √13
√
b) − 3
c) -1
Using trigonometric identities practice (page 48)
Q61:
RHS = tan a sin a
sin a
× sin a
=
cos a
sin2 a
if sin2 x + cos2 x = 1 then sin2 x = 1 − cos2 x
=
cos a
1 − cos2 a
=
cos a
= LHS
Using trigonometric identities exercise (page 48)
Q62:
a) 1 − sin2 x
b) 1 − cos2 x
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ANSWERS: TOPIC 21
c) cos x tan x
d)
sin x
tan x
Q63:
a) 2
b) tan A
Q64:
a) sin2 x
b) 2 cos A
Q65:
a)
RHS = (sin X + cos X)2
= sin X 2 + 2 sin X cos X + cos X 2
= sin X 2 + cos X 2 + 2 sin X cos X
= 1 + 2 sin X cos X
= LHS
b)
RHS = (sin x − cos x)2 + (sin x + cos x)2
= sin x2 − 2 sin x cos x + cos x2 + sin x2 + 2 sin x cos x + cos x2
= sin x2 + cos x2 + sin x2 + cos x2 − 2 sin x cos x + 2 sin x cos x
= 1 + 1
= 2
= LHS
c)
RHS = cos2 A − sin2 A
= cos2 A − 1 − cos2 A
= cos2 A − 1 + cos2 A
= 2cos2 A − 1
= LHS
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ANSWERS: TOPIC 21
End of topic 21 test (page 52)
Q66:
Q67:
Q68:
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ANSWERS: TOPIC 21
Q69:
Q70:
Q71:
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ANSWERS: TOPIC 21
Q72:
Q73:
Q74:
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ANSWERS: TOPIC 21
Q75:
Q76:
Q77:
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ANSWERS: TOPIC 21
Q78:
Q79:
Q80:
Q81:
a) y = sin 1 · 5x◦
b) 1·5
Q82:
a) y = cos 0 · 5x◦
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ANSWERS: TOPIC 21
b) 0·5
Q83:
a) y = 4 cos 3 · 5x◦
b) 4
Q84:
a) y = 3 sin 3 · 5x◦
b) 3
Q85:
a) y = 4 cos 3 · 5x◦ − 1
b) 4
Q86:
a) y = − 2 sin 4x◦ − 2
b) -2
Q87:
a) y = − 3 tan 2 · 5x◦ − 2
b) -3
Q88: y = − 4 cos (x◦ + 60)
Q89: y = 4 tan (x◦ − 30)
Q90: y = 3 sin (x◦ + 15)
Q91: y = − 2 tan (x◦ − 30) + 2
Q92: y = 4 sin (x◦ − 15) − 1
Q93: y = − 2 cos (x◦ − 30) − 2
Q94: x = 10 · 4 and x = 169 · 6
Q95:
Hint
•
Put this in your calculator to help you find 2 solutions remembering to use a
quadrant chart.
Answer: x = 66 · 4 and x = 293 · 6
Q96:
Hint
•
Put this in your calculator to help you find 2 solutions remembering to use a
quadrant chart.
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ANSWERS: TOPIC 21
Answer: x = 18 · 4 and x = 198 · 4
Q97:
Hint
•
Put the positive value in your calculator to help you find 2 solutions remembering
to use a quadrant chart.
Answer: x = 194 · 5 and x = 345 · 5
Q98:
Hint
•
Put the positive value in your calculator to help you find 2 solutions remembering
to use a quadrant chart.
Answer: x = 135 · 6 and x = 224 · 4
Q99: 1/2
Q100:
Q101:
√1
2
√
3
Q102:
Steps:
•
135◦ is in the sin quadrant, what is the associated acute angle (Hint: 180 - a)? 45
•
Use this to help you find the exact value.
Answer: − √13
Q103:
Steps:
•
330◦ is in the cos quadrant, what is the associated acute angle (Hint: 360 - a)? 30
•
Use this to help you find the exact value.
Answer:
√1
2
Q104:
Steps:
•
120◦ is in the tan quadrant ,what is the associated acute angle (Hint: 180 + a)? 60
•
Use this to help you find the exact value.
Answer: −1/2
Q105:
2 cos2 Y + 2 sin2 Y
= 2 cos2 Y + sin2 Y
= 2 × 1
= 2
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ANSWERS: TOPIC 21
Q106:
sin A × sin A × sin A
sin3 A
=
3
cos A
cos A × cos A × cos A
= tan A × tan A × tan A
= tan3 A
Q107:
4 cos x + 5 sin x = 0
sin x
0
cos x
+ 5
=
4
cos x
cos x
cos x
4 + 5 tan x = 0
5 tan x = −4
4
tan x = −
5
4
When tan x = 5 , x = 38 · 7
So x = 180 − 38 · 7 = 141 · 3 and x = 360 − 38 · 7 = 321 · 3
Q108:
RHS = (3 cos x + 2 sin x)2 + (2 cos x − 3 sin x)2
= 9 cos2 x + 12 cos x sin x + 4 sin2 x + 4 cos2 x − 12 cos x sin x + 9 sin2 x
= 13 cos2 x + 13 sin2 x
= 13 cos2 x + sin2 x
= 13 × 1
= 13
= LHS
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