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5
Trigonometry
The Canadarm 2 robotic manipulator is a
robotic system which has been operating on
the International Space Station since 2001.
Its primary uses are moving equipment and
supporting astronauts whilst they are working
on the exterior of the station. The machinery is
controlled by changing the angles of its joints,
which requires repeated use of trigonometric
functions.
Objectives
Sketch and use graphs of the sine, cosine and tangent functions (for angles of any size and
using either degrees or radians).
Use the exact values of the sine, cosine and tangent of 30°, 45°, 60° and related angles,
e.g. cos 150° = 3 .
2
Use the notations sin−1 x, cos−1 x and tan−1 x to denote the principal values of inverse
trigonometric relations.
Use the identities sinT { tanT and sin2 θ + cos2 θ ≡ 1.
Find all the solutions of simple trigonometric equations lying in a specified interval.
cosT
Before you start
You should know how to:
Skills check:
1. Use the trigonometric ratios and Pythagoras’
rule for a right-angled triangle.
1. Find the side or angle marked x.
2
2
2
e.g. a) y = 5 + 7
y
y = 74 = 8.60 cm (3 sf)
b) tan x = 5 x = 35.5° (1 dp)
a)
7 cm
35°
b)
x
8 mm
6 mm
5 cm
c)
8m
x
3m
x
x
7 cm
7
2. Convert between degrees and radians as an
angle of measure.
e.g. 75° = S × 75 = S = 1.31 rad.
180
12
2. Convert to radians
i) 90°
ii) 225°
Convert to degrees
S
ii)
rad
i) 3S rad
4
5
iii) 43°
iii) 2.5 rad
71
3. Solve quadratic equations.
3. Solve
2
a) 2x2 – 7x – 4 = 0
a) x – 4x –21 = 0
b) x2 – 5x + 2 = 0
(x + 3) (x – 7) = 0 x = –3 or 7
b) 3x2 + x – 5 = 0
x=
1 r 12 4 u 3 u 5
x=
6
1 r 61
6
= –1.47 or 1.14 (3 sf)
5.1 Exact values of trigonometric functions
y
The unit circle can be used to define the trigonometric ratios.
The angle θ is measured in an anticlockwise direction from the
positive x-axis
1
P (cos i, sin i)
1
cosθ can be defined as the x coordinate of P.
sinθ can be defined as the y coordinate of P.
sin i
i
–1
0
cos i
1
x
tanθ can be defined as sinT .
cosT
–1
The diagram can be used to write down the values of sin, cos and tan for
0°, 90°, 180°, 270° and 360°.
Example 1
Without using a calculator, write down the values of
a) cos0°
b) sin90°
c) cos180°
e) tan0°
f) tan90°.
a) cos0° = 1
cos0° is the x coordinate of the point P when θ = 0°
b) sin90° = 1
sin90° is the y coordinate of the point P when θ = 90°
c) cos180° = −1
cos180° is the x coordinate of the point P when θ = 180°
d) cos270° = 0
e) tan0° = sin 0
cos 0
f) tan90° = sin 90
cos 90
cos270° is the x coordinate of the point P when θ = 270°
0=0
1
1
0
tan90° is undefined
(or infinitely large)
72
d) cos270°
Exact values of trigonometric functions
Use tanθ =
sinT
cosT
Division by 0 is undefined
Some exact values of sin, cos and tan can be found easily from the triangles below,
and are also shown in the table.
Degrees
Radians
sin
cos
tan
30°
S
6
1
2
3
2
1
3
45°
S
4
1
2
1
2
1
60°
S
3
3
2
1
2
30° 30°
45°
√2
2
2
√3
1
60°
45°
60°
1
1
1
The sin, cos and tan of other angles in the range 0° to 360° can be found.
For every angle between 0° and 90° there is a related angle in
each of the other three quadrants. These related angles give
the same numerical value for each trigonometric function
but the signs change.
For example, the angles related to 30° are
180° − 30° = 150°
180° + 30° = 210°
360° − 30° = 330°
3
Remember that angles are
measured anticlockwise
from the positive x-axis.
90°
180°
30°
30°
30°
30°
This diagram shows which functions are positive in
each quadrant.
0°
270°
We can write down the sin, cos and tan for related angles, so
sin 150° = sin 30° = 1
Sine
All
Tangent
Cosine
2
sin 210° = −sin 30° = − 1
2
sin 330° = −sin 30° = − 1
2
cos 150° = −cos 30° = − 3
2
Use the table of values
above and the diagram to
help you.
cos 210° = −cos 30° = − 3
2
Trigonometry
73
cos 330° = cos 30° = 3
2
tan 150° = −tan 30° = − 1
3
tan 210° = tan 30° = 1
3
tan 330° = −tan 30° = − 1
3
Example 2
Find the exact values of
a) sin 120°
b) cos 120°
c) tan 120°
a) sin 120° = sin 60°
P
= 3
e) cos 225°
f) tan 300°.
y
120°
2
d) sin 225°
Find the y coordinate of P when θ = 120°, which
is equal to the y coordinate when θ = 60°.
60°
x
0
Find the x coordinate of P when θ = 120°, which
is the negative of the x coordinate when θ = 60°.
b) cos 120° = −cos 60°
= −1
2
c) tan 120° = −tan 60°
When θ = 120°, sin θ > 0, cos θ < 0 so tan θ < 0
y
=− 3
225°
d) sin 225° = −sin 45°
=− 1
2
x
Use the angle with the (negative) x-axis.
x
Use the angle with the (positive) x-axis.
e) cos 225° = −cos 45°
=− 1
y
2
300°
f) tan 300° = −tan 60°
=− 3
Exercise 5.1
1. Find the exact values of
a) cos 240°
b) tan 135°
c) sin 300°
d) cos 315°.
c) sin 3S
d) tan 5S .
2. Find the exact values of
a) sin 3S
4
74
b) sin 4S
3
Exact values of trigonometric functions
2
3
3. Without using a calculator, state whether each of these is positive or negative.
a) sin 130°
b) cos 130°
c) sin 255°
d) cos 255°
g) sin 255°
h) cos 255°.
Use your calculator to find the value of
e) sin 130°
f) cos 130°
4. Express the following in terms of the related acute angle.
a) sin 132°
b) cos 310°
c) tan 215°
d) sin 220°
e) cos 153°
f) tan 148°
g) cos 195°
h) sin 335°
5. a) Use your calculator to find the angle (to the nearest degree) between
0° and 90° whose sine is 0.36.
b) Hence find another angle between 0° and 360° whose sine is 0.36.
6. Find two angles (to the nearest degree) in the range 0° < x° < 360° such
that cos x° = 0.3.
7. Find all the angles (to the nearest 0.1°) between 0° and 360° whose
cosine is 0.7660.
8. Find all the angles (to the nearest 0.1°) between 0° and 360° whose sine
is −0.3636.
9. Solve each of these equations, giving all solutions between 0° and 360°
to the nearest degree.
a) sin x° = 0.9
b) cos x° = 0.9
c) sin x° = −0.6
d) cos x° = 0.33
e) tan x° = 0.25
f) tan x° = −0.44
10. Solve each of these equations, giving all solutions between 0 and 2π in
radians (correct to 3 s.f.).
a) sin x = 0.4
b) cos x = 0.4
c) sin x = −0.8
d) cos x = −0.21
e) tan x = 0.75
f) tan x = −0.36
5.2 Graphs of trigonometric functions
In section 5.1 we saw how to define sin, cos and tan using the unit circle.
These definitions apply for all values of θ, both positive (anticlockwise)
and negative (clockwise).
An angle outside the interval 0° ≤ x° < 360° has an equivalent angle in the
interval 0° ≤ x° < 360°.
For example,
and
an angle of 420° is equivalent to an angle of 60°
an angle of −120° is equivalent to an angle of 240°.
Trigonometry
75
The graph of y = sin x shows that the curve is infinite but repeats every 360° (or 2π radians).
We say y = sin x is periodic and has a period of 360° (or 2π radians).
y = sin x
1
–x°
–360° –270° –180° –90°
0
90°
–1
180°
270°
450°
540°
630°
720°
x
360 + x°
180 – x°
x°
360°
The graph can
be drawn using
degrees or radians
From the graph we can write down some relationships.
For example, sin (−x°) = −sin x°
sin (180 − x)° = sin x°
sin (360 + x)° = sin x°
These equivalences can
also be seen from the unit
circle shown in section 5.1.
The graph of y = cos x shown below behaves in a similar way.
y = cos x
1
–360° –270° –180° –90°
180 – x°
0
–1
–x°
90°
x°
180°
270°
360°
360 – x°
450°
540°
630°
720°
x
360 + x°
cos (−x°) = cos x°
cos (180 − x)° = −cos x°
cos (360 − x)° = cos x°
cos (360 + x)° = cos x°
We have defined tan x as sin x .
cos x
There is a problem when x = 90°, 270°, … because cos x° = 0 at these points
and so the value of tan x is undefined (or infinite) at these points. There are
asymptotes at these values of x on the graph of y = tan x.
y = tan x
–360° –270° –180° –90°
0
90° 180° 270° 360° 450° 540° 630° 720°
x
The graph of y = tan x is periodic and has a period of 180° (or π radians).
Examples of relationships for this function are
tan (−x°) = −tan x°
tan (90 + x)° = −tan (90 − x)°
tan (180 + x)° = tan x°
76
Graphs of trigonometric functions
Use the graph to check you
understand why
The graphs of sine, cosine and tangent can be used to help you find all the
angles, x, within a given interval for which sin x = k or cos x = k (−1 < k < 1)
or tan x = k (k any real number).
From the sine and cosine graphs we see that there are two angles, x,
between 0° and 360° for each value of k. Further angles can be found by
taking each of these two values and adding or subtracting multiples of 360°.
For the tangent graph we see that there is one angle,
x, between −90° and 90° for each value of k.
Further angles can be found by taking this value
and adding or subtracting multiples of 180°.
Note: a calculator will give you only one
angle (called the principal value), you can
sketch the graph to find other angles.
Radians may be used instead of degrees.
Example 3
Find all the angles x° (to the nearest degree), where 0° < x° < 360°, such that sin x° = 0.88
x = sin−1 (0.88)
= 62°
x = 62°, 180° − 62°
x = 62°, 118°
Use a calculator to find the principal value
Use a graph sketch to find the second angle in
the stated interval
Example 4
Solve tan x° = 2.35 for 0° < x° < 720°.
Give your answers correct to one decimal place.
x = tan−1 (2.35)
= 66.9°
This is the principal value given by a calculator
x = 66.9°, 180° + 66.9°, 360° + 66.9°, 540° + 66.9°
x = 66.9°, 246.9°, 426.9°, 606.9°
Add on multiples of 180° to give all values in
the interval 0° < x° < 720°
Exercise 5.2
1. sin 40° = 0.643 (3 s.f.)
a) Write down another angle between 0° and 360° whose sine is 0.643
b) Write down two angles between 360° and 720° whose sine is 0.643
2. Given tan S = 1, write down all the angles between 0 and 6π whose
4
tangent is 1.
Use graph sketches in
each question to help
you find the angles
required.
3. Find all the angles between 0° and 360° whose cosine is −0.766.
4. Find all the angles between 0° and 360° whose sine is −0.25.
Trigonometry
77
5. Solve each of these equations, giving all solutions between 0° and 360°.
a) sin x° = 0.384 b) tan x° = 1.988 c) cos x° = 0.379 d) sin x° = −0.2
6. Solve each of these equations, giving all solutions between 0 and 4π.
a) sin x = 1
2
b) cos θ = 1
2
c) sin x = −1
d) tan θ = −1
7. a) Express sin (180 + x)° in terms of sin x.
b) Express cos (180 − x)° in terms of cos x.
c) Express tan (180 + x)° in terms of tan x.
d) Express tan (360 − x)° in terms of tan x.
8. a) Express cos (3π + x) in terms of cos x.
c) Express sin (x − π) in terms of sin x.
b) Express sin (x + 4π) in terms of sin x.
d) Express tan (x − π) in terms of tan x.
9. Solve the equation 3sin x = 2cos x, giving all solutions between 0° and 360°.
Use tan x = sin x
cos x
5.3 Composite graphs
The graph of y = sin x° has period 360°.
The range of values for y = sin x° is −1 ≤ y ≤ 1
The graph of y = asin x° has period 360°.
It is a stretch of y = sin x°, factor a, parallel to the y-axis.
The range of values for y = asin x° is −a ≤ y ≤ a
y
y = asinx°
a
–y = asinx
1
–y = sinx
–180° –90° 0
–1
90° 180° 270° 360° x
–a
The graph of y = sin bx° has period 360q .
b
It is a stretch of y = sin x°, factor 1 , parallel to the x-axis.
y
1
b
The range of values for y = sin bx° is −1 ≤ y ≤ 1
The graph of y = sin (x + c)° has period 360°.
§ c ·
It is a translation of y = sin x° by ¨ ¸.
© 0¹
The range of values for y = sin (x + c)° is −1 ≤ y ≤ 1
78
Composite graphs
–180° –90° 0
–1
y = sin bx°
360°
b
90° 180° 270° 360° x
y
1
–c°
–180° –90° 0
–1
y = sin (x + c)°
90° 180° 270° 360° x
The graph of y = sin x° + d has period 360°.
§0·
It is a translation of y = sin x° by ¨ ¸.
©d ¹
The range of values for y = sin x° + d is −1 + d ≤ y ≤ 1 + d.
y
1+d
y = sin x° + d
1
–180° –90°
The graphs of y = cos x° and y = tan x° can be transformed
in the same way.
0
–1
90° 180° 270° 360°
x
Example 5
Sketch the graph of y = 3cos x° for 0° ≤ x ≤ 360°.
y
3
2
1
y = 3cos x
0
90° 180° 270° 360°
x
–1
–2
Stretch the graph of y = cos x° parallel to the y-axis, factor 3
–3
Example 6
Sketch the graph of y = sin 2x for 0 ≤ x ≤ 2π.
y
y = sin 2x
1
0
–1
r
2
r
Stretch the graph of y = sin x parallel to the x-axis, factor
3r
2
2r
1
2
x
Use radians to label the x-axis
Exercise 5.3
1. What is the period of each of these functions
a) y = sin 2x°
b) y = cos 5x°
c) y = cos x° + 1
d) y = 5 sin (x − 30)°
e) y = tan 1 x°
2
f) y = 2 tan 3x° − 4
g) y = tan (x + 45)°
h) y = 3 sin (2x − 60)°.
Trigonometry
79
2. Sketch, on separate diagrams, the graphs of
a) y = sin 3x° for 0° ≤ x ≤ 360°
b) y = −cos 2x° for 0° ≤ x ≤ 360°
c) y = tan 1 x° for −360° ≤ x ≤ 360°
2
d) y = 2 sin 4x° for 0° ≤ x ≤ 180°
e) y = 3 tan (x + 30)° for 0° ≤ x ≤ 360°
f) y = 3 sin 2x° − 1 for 0° ≤ x ≤ 360°.
3. Sketch, on separate diagrams, the graphs of the following functions
for 0 ≤ θ ≤ 2π.
b) y = tan T S
a) y = 6 cos θ + 2
c) y = sin T S
2
d)
4
y = cos 2T S 2
4. Find an equation for each graph.
a)
b)
y
1
1
0
90° 180° 270° 360°
0
x
–1
c)
y
90° 180° 270° 360°
x
90° 180° 270° 360°
x
–1
y
d)
y
1
4
0
2
–1
0
–2
90° 180°270°360° x
–4
–2
–3
5. a) Sketch on the same diagram, the graphs of y = sin 2θ and,
y = cos θ for 0° ≤ θ ≤ 180°.
b) State the number of roots of the equation sin 2θ = cos θ for
which 0° ≤ θ ≤ 180°.
c) Find the roots of the equation sin 2θ = cos θ for which 0° ≤ θ ≤ 360°.
80
Composite graphs