Download Trigonometric rules, graphs and identities

SCHOLAR Study Guide
National 5 Mathematics
Assessment Practice
Topic 7:
Trigonometric
graphs and identities
Authored by:
Margaret Ferguson
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
rules,
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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SCHOLAR Study Guide Assessment Practice Topic 7: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
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1
Topic 1
Trigonometric rules, graphs and
identities
Contents
7.1
Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
7.2
Assessment practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2
TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
By the end of this topic, you should have identified your strengths and areas for further
revision. Read through the learning points before you attempt the assessments and go
back to the Course Materials unit if you need more help.
You should be able to:
•
use the:
◦
area of a triangle rule;
◦
Sine rule for a side;
◦
Sine rule for an angle;
◦
Cosine rule for a side;
◦
Cosine rule for an angle;
•
choose the most appropriate rule to use;
•
use bearings to find a distance or direction;
•
identify the:
•
◦
features of Sine, Cosine and Tangent graphs;
◦
amplitude and period of a trig function;
identify and sketch trig graphs with:
◦
multiple angles;
◦
vertical translations;
◦
horizontal translations;
•
solve trig equations;
•
determine exact trigonometric values;
•
use trig identities to simplify trig expressions.
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
1.1
Learning points
Trigonometric rules
Area Rule
1
2 ab
•
Area =
sin C
•
Two sides and the angle in-between are required.
Sine Rule for a side
•
a
sin A
•
Two angles and the side opposite one of the angles are required.
=
b
sin B
=
c
sin C
Sine Rule for an angle
•
a
sin A
•
Two sides and the angle opposite one of the sides are required.
=
b
sin B
=
c
sin C
Cosine rule for a side
•
a2 = b2 + c2 − 2bc cos A
•
Two sides and the angle in-between are required.
Cosine rule for an angle
b2 + c2 − a2
2bc
•
cos A =
•
All three sides are required.
Bearings
•
A bearing is measured clockwise from a north line and has three digits.
•
You may have to use angle properties:
◦
Supplementary angles = 180 ◦ .
◦
Angles round a point = 360 ◦ .
◦
The sum of the angles in a triangle = 180 ◦ .
◦
Alternate angles or "Z" angles are equal.
◦
Corresponding angles or "F" angles are equal.
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4
TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Trigonometric graphs and identities
Sketching and identifying trigonometric graphs
•
The graph of y = sin x looks like this:
1
0
90
180
270
360
-1
•
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;
c has:
b) +
c has:
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
◦
5
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◦
1
2
√
3
2
√1
2
√
3
2
√1
2
1
2
√1
3
Or learn how to draw the 2 special triangles.
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1
60◦
√
3
90◦
1
0
undefined
6
TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
•
•
Trigonometric identities
•
sin2 x + cos2 x = 1
•
tan x =
sin x
cos x
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
1.2
7
Assessment practice
Make sure that you have read through the learning points or completed some revision
before attempting these questions. Tailor your practice by choosing the most appropriate
questions.
•
Trig Rules Questions 1 - 13
•
Trig Graphs Questions 14 - 21
•
Trig Equations, Exact Values and Identities Questions 22 - 56
Key point
Questions 6, 7, 8, 9 10, 11, 12, 13, 53, 54, 55 and 56 also assess your reasoning
skills.
Assessment practice: Trigonometric rules, graphs and identities
Go online
Trigonometric Rules
Q1: Calculate the area of the triangle.
..........................................
Q2: Calculate the size of angle A.
..........................................
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Q3:
Calculate the length of b.
..........................................
Q4:
Calculate the length of c.
..........................................
Q5:
What is the size of angle D?
..........................................
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Garry’s back garden is a quadrilateral as shown below.
He was able to measure some sides and angles.
Q6: Calculate the length of AC.
..........................................
Q7: Calculate the size of ∠BAC.
..........................................
Q8: Calculate the length of CD.
..........................................
Q9: What is the size of ∠ACD
..........................................
Q10: Calculate the area of the garden.
..........................................
The warship (W ) is on a bearing of 125 ◦ from the lighthouse (L).
The boat (B) is on a bearing of 228 ◦ from the lighthouse.
Q11: What is the size of ∠BLW ?
..........................................
Q12:
The warship and the boat are 32 miles apart and the warship is 21 miles from the
lighthouse.
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Calculate the size of ∠LBW .
..........................................
Q13: What is the bearing of the warship from the boat?
..........................................
Trigonometric Graphs
Q14: Sketch the graph of y = sin 4x for 0 ≤ x ≤ 360.
..........................................
Q15: Sketch the graph of y = 5 cos 4x − 3 for 0 ≤ x ≤ 360.
..........................................
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Q16: Sketch the graph of y = tan(x − 45) + 1 for 0 ≤ x ≤ 360.
..........................................
Here is a sketch of the graph y = a sin bx, 0 ≤ x ≤ 360.
Q17: What is the equation of the graph?
..........................................
Q18: What is the amplitude?
..........................................
Q19: What is the period?
..........................................
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Q20: Here is a sketch of the graph y = a sin bx + c, 0 ≤ x ≤ 360.
What is the equation of this graph?
..........................................
Q21: Here is a sketch of the graph y = a cos (x + b) + c, 0 ≤ x ≤ 360.
What is the equation of this graph?
..........................................
Solving Trigonometric Equations
Q22: Solve sin x = 0 · 58
..........................................
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Q23: Solve 5 cos x = 3
..........................................
Q24: Solve 2 tan x − 9 = 0
..........................................
Q25: Solve 6 sin x = − 1
..........................................
Q26: Solve 9 cos x + 7 = 0
..........................................
Exact Trigonometric Values
Do not use a calculator for these questions.
Q27: What is the exact value of sin 30 ◦ ?
..........................................
Q28: What is the exact value of cos 45◦ ?
..........................................
Q29: What is the exact value of tan 30◦ ?
..........................................
Q30: What is the exact value of cos 60◦ ?
..........................................
Q31: What is the exact value of tan 45◦ ?
..........................................
Q32: What is the exact value of sin 60◦ ?
..........................................
Q33: What is the exact value of sin 120◦ ?
..........................................
Q34: What is the exact value of cos 330◦ ?
..........................................
Q35: What is the exact value of tan 210◦ ?
..........................................
Q36: What is the exact value of cos 315◦ ?
..........................................
Q37: What is the exact value of tan 240◦ ?
..........................................
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14
TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Q38: What is the exact value of sin 135◦ ?
..........................................
Q39: What is the exact value of tan 225◦ ?
..........................................
Q40: What is the exact value of sin 150◦ ?
..........................................
Q41: What is the exact value of cos 300◦ ?
..........................................
Q42: What is the exact value of sin 210◦ ?
..........................................
Q43: What is the exact value of cos 120◦ ?
..........................................
Q44: What is the exact value of tan 135◦ ?
..........................................
Q45: What is the exact value of tan 300◦ ?
..........................................
Q46: What is the exact value of sin 225◦ ?
..........................................
Q47: What is the exact value of cos 210◦ ?
..........................................
Q48: What is the exact value of cos 225◦ ?
..........................................
Q49: What is the exact value of sin 330◦ ?
..........................................
Q50: What is the exact value of tan 330◦ ?
..........................................
Q51: What is the exact value of sin 315◦ ?
..........................................
Q52: What is the exact value of cos 150◦ ?
..........................................
Q53: What is the exact value of tan 150◦ ?
..........................................
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TOPIC 1. TRIGONOMETRIC RULES, GRAPHS AND IDENTITIES
Trigonometric Identities
Q54: Simplify 9sin2 A + 9cos2 A
..........................................
Q55: Simplify
1 − cos2 Y
cos2 Y
..........................................
Q56: Solve 6 cos x + 5 sin x = 0
..........................................
Q57: Prove that tan2 θ − tan2 θsin2 θ = sin2 θ
..........................................
..........................................
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16
ANSWERS: TOPIC 7
Answers to questions and activities
7 Trigonometric rules, graphs and identities
Assessment practice: Trigonometric rules, graphs and identities (page 7)
Q1:
793 · 4 cm2
Q2:
Steps:
•
What rule would you use when you have 2 sides and 1 angle? sine
•
Use this rule to find the angle A.
Answer: 28◦
Q3:
Steps:
•
What rule would you use when you have 2 sides and the angle between them?
cosine
•
Use this rule to find the side b.
Answer: 28 · 5 m
Q4:
Steps:
•
What rule would you use when you have 2 angles and 1 side? sine
•
Use this rule to find the side c.
Answer: 2 · 2 km
Q5:
Steps:
•
What rule would you use when you have all 3 sides? cosine
•
Use this rule to find angle D.
Answer: 60 · 8◦
Q6:
Steps:
•
Look at triangle ABC.
•
What rule would you use when you have 2 sides and the angle between them?
cosine
•
Use this rule to find the side AC.
Answer: 17 · 9 m
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ANSWERS: TOPIC 7
Q7:
Steps:
•
Look at triangle ABC.
•
What rule would you use when you have all 3 sides? cosine
•
Use this rule to find ∠BAC.
Answer: 41◦
Q8:
Steps:
•
Look at triangle ACD.
•
What rule would you use when you have 2 angles and 1 side? sine
•
Use this rule to find the side CD.
Answer: 12 · 4 m
Q9:
Hint:
•
The sum of the angles in any triangle = 180 ◦
Answer: 58◦
Q10:
Steps:
•
What is the area of triangle ABC? 64 · 6 m 2
•
What is the area of triangle ACD? 94 · 1 m 2
•
Use your answers to find the area of the whole garden
Answer: 158 · 7 m2
Q11: 103◦
Q12:
Steps:
•
What rule would you use when you have 2 sides and 1 angle? sine
•
Use this rule to find ∠LBW .
Answer: 39 · 7◦
Q13:
Steps:
•
What is the size of ∠BLN2 ? 132◦
•
What is the supplement of ∠BLN 2 ? 48◦
•
Use this answer to find the 3 figure bearing of the warship from the boat.
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17
18
ANSWERS: TOPIC 7
Answer: 088◦
Note: The 0 is essential when giving a bearing as an answer.
Q14:
Q15:
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ANSWERS: TOPIC 7
Q16:
Q17: y = 5 sin 3x
Q18: 5
Q19: 120
Q20: y = 2 sin 4x − 3
Q21: y = 3 cos(x + 45◦ ) + 2
Q22: x = 35 · 5 and x = 144 · 5
Q23:
Steps:
3
5
•
What is cos x?
•
Put this in your calculator to help you find 2 solutions remembering to use a
quadrant chart.
Answer: x = 53 · 1 and x = 306 · 9
Q24:
Steps:
9
2
•
What is tan x?
•
Put this in your calculator to help you find 2 solutions remembering to use a
quadrant chart.
Answer: x = 77 · 5 and x = 257 · 5
Q25:
Steps:
•
What is sin x? − 16
•
Put the positive value in your calculator to help you find 2 solutions remembering
to use a quadrant chart.
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19
20
ANSWERS: TOPIC 7
Answer: x = 189 · 6 and x = 350 · 4
Q26:
Steps:
•
What is cos x? − 79
•
Put the positive value in your calculator to help you find 2 solutions remembering
to use a quadrant chart.
Answer: x = 141 · 1 and x = 218 · 9
Q27:
1
2
Q28:
√1
2
Q29:
√1
3
Q30:
1
2
Q31: 1
√
3
2
Q32:
Q33:
Steps:
•
Is sin 120◦ positive or negative? positive
•
What is the associated acute angle? 60 ◦
•
Use these answers to help you find the exact value.
√
Answer:
3
2
Q34:
Steps:
•
Is cos 330◦ positive or negative? positive
•
What is the associated acute angle? 30 ◦
•
Use these answers to help you find the exact value.
√
Answer:
3
2
Q35:
Steps:
•
Is tan 210◦ positive or negative? positive
•
What is the associated acute angle? 30 ◦
•
Use these answers to help you find the exact value.
Answer:
Q36:
√1
3
√1
2
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ANSWERS: TOPIC 7
Q37:
Q38:
√
3
√1
2
Q39: 1
Q40:
1
2
Q41:
1
2
Q42:
Steps:
•
Is sin 210◦ positive or negative? negative
•
What is the associated acute angle? 30 ◦
•
Use these answers to help you find the exact value.
Answer: − 12
Q43:
Steps:
•
Is cos 120◦ positive or negative? negative
•
What is the associated acute angle? 60 ◦
•
Use these answers to help you find the exact value.
Answer: − 12
Q44:
Steps:
•
Is tan 135◦ positive or negative? negative
•
What is the associated acute angle? 45 ◦
•
Use these answers to help you find the exact value.
Answer: −1
√
Q45: − 3
Q46: − √12
Q47: −
√
3
2
Q48: − √12
Q49: − 12
Q50: − √13
Q51: − √12
Q52: −
√
3
2
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22
ANSWERS: TOPIC 7
Q53: − √13
Q54:
Steps:
•
Factorise 9sin2 A + 9cos2 A. 9(sin2 A + cos2 A)
•
Use one of the trig identities to simplify fully.
Answer: 9
Q55:
Steps:
•
Simplify 1 − cos2 Y . sin2 Y
•
Use one of the trig identities to simplify fully.
Answer: tan2 Y
Q56:
Hint:
•
Divide each term by cos x and use one of the trig identities then solve the trig
equation as normal.
Steps:
•
What is the simplified equation? 6 + 5 tan x = 0
•
Make tan x the subject of the formula, what is tan x? − 65
•
Put the positive value in your calculator to help you find 2 solutions remembering
to use a quadrant chart.
Answer: x = 129 · 8 and x = 309 · 8
Q57:
LHS = tan2 θ − tan2 θsin2 θ
= tan2 θ(1 − sin2 θ)
= tan2 θ × cos2 θ
sin2 θ
× cos2 θ
cos2 θ
= RHS
=
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