MathsStart
(NOTE Feb 2013: This is the old version of MathsStart.
New books will be created during 2013 and 2014)
Topic 6
Trigonometry II
y y
P
1
O x
Q – 90° 90°
negative positive
MATHS LEARNING CENTRE
Level 3, Hub Central, North Terrace Campus
The University of Adelaide, SA, 5005
TEL 8313 5862 | FAX 8313 7034 | EMAIL mathslearning@adelaide.edu.au
www.adelaide.edu.au/mathslearning/
This Topic...
This topic introduces a new way of thinking about angles, and extends the definitions of
sine, cosine and tangent to angles greater than 90°. It explores the properties and graphs
of the trigonometric functions sin , cos and tan , and their applications.
Author of Topic 6: Paul Andrew.
__ Prerequisites ______________________________________________
You will need a scientific calculator.
We also assume you have read Topic 5: Trigonometry I
__ Contents __________________________________________________
Chapter 1 Measuring Angles.
Chapter 2 Sine, Cosine and Tangent Functions.
Chapter 3 Applications.
Chapter 4 Identities.
Chapter 5 Radian Measure.
Appendices
A. Answers
Printed: 18/02/2013
1
Measuring Angles
In module 1, we saw how the counting numbers N were extended to the real numbers R. It
was necessary to invent new numbers because N was not closed under subtraction, for
example the calculation 3 – 7 did not have an answer among the natural. We also saw how
negative numbers only gained acceptance after they were freshly interpreted as being points
on a number line, with the numbers on the right of the origin 0 being taken as positive and
numbers on the left being taken as negative.
A similar problem occurs with angles. In Module 5, we calculated and interpreted angles like
30° + 30° and 90° – 45°, but what meaning should we give to the ‘angle’ 30° – 60°?
Angles can be freshly interpreted in a coordinate plane, as being measured from the positive
x-axis. Angles measured in an anti-clockwise direction are thought of as positive, and angles
measured in a clockwise direction as thought of as being negative. This seems very different
to how we have been thinking about angles inside triangles, but you will find later that both
interpretations give the same answers.
y y
positive
O x O x
negative
Example
Sketch the angles 30° and –60°.
Answer Divide the right-angle into
three equals parts to sketch
30° and 60°.
30°
–60°
1
Sine, Cosine and Tangent 2
Example
Sketch the angle 405° = 360° + 45°
Answer
360° + 45° is one revolution
plus an additional 45°.
1. Problems
Sketch the following angles
(a) ±120° (b) ±135° (c) ±150° (d) ±210° (e) ±225°
(f) ±240° (g) ±300° (h) ±315° (i) ±330° (j) ±390°
2
Sine, Cosine and Tangent
2.1 The Unit Circle.
The distance between two points (x0, y0) and (x1, y1) in the coordinate plane is
distance = (x1 x0 )2 (y1 y0 )2
This can be deducedfrom the diagram below using Pythagoras’ Theorem, and is true
wherever the points (x0, y0) and (x1, y1) are in the plane.
y
(x1, y1)
rise = y1 – y0
(x0, y0)
run = x1 – x0
The unit circle is a circle with centre at the origin (0, 0) and radius 1. As the distance of each
point (x, y) on the circle from the origin is equal to 1, the coordinates of the points on the unit
circle satisfy the equation (x 0)2 (y 0)2 = 1 or x 2 y2 = 1.
y
• (x, y)
1
(0, 0) x
3
Sine, Cosine and Tangent 4
2.2 The Sine and Cosine Functions
In Topic 5, we saw that (sin ) 2 (cos ) 2 1 for any angle in a right-angled triangle. The
diagram below shows that this is another way of interpreting Pythagoras’ Theorem.
1
sin
cos
You can see that if P is a point in the first quadrant of the unit circle, and if the angle between
OP and the positive x-axis is , then P must have coordinates (cos, sin).
• P (cos, sin)
1
O
This is only true when is an angle in a right-angled triangle, ie. 0° < < 90°, because sin
and cos are only defined for angles in right-angled triangles. However . . . we can use this
idea to define sin and cos for any angle .
We define sin and cos for any angle as being the coordinates of the point P on the unit
circle, when OP has angle with the positive x-axis. Remember: positive angles are measured
in an anticlockwise direction and negative angles are measured in a clockwise direction.
Example
What are the exact values of sin 135°, cos 135°, sin (–45°) and cos (–45°)?
Answer
1 1
When = 45°, sin and cos , so the point on the unit circle corresponding to the
2 2
1 1
angle 45° is , .
2 2
5 Trigonometry II
1 1 1 1
2
, •
2
• 2 ,
2
45°
135°
O –45°
1
1 1
• ,
2 2
1 1 1 1
So sin135 , cos135 , sin(45) and cos(45) .
2 2 2 2
The word ‘function’ is used
here as it doesn’t make much
sense to say ‘ratios’ any more
Problems 2.2
Find the exact values of the trigonometric functions below, and check you answers using a
calculator.
(a) sin 120° and cos 120° (b) sin 150° and cos 150°
(c) sin 210° and cos 210° (d) sin 330° and cos 330°
(e) sin(–30°) and cos(–30°) (f) sin(–150°) and cos(–150°)
2.3 Graphs of the Sine and Cosine Functions
The unit circle can be used to draw the graph of y sin , using the idea that the length of the
line PQ is equal to sin .
y y
P
1
O x
Q – 90° 90°
negative positive
Sine, Cosine and Tangent 6
In the diagram above, you can see that as increases from 0° to 90° (in an anticlockwise
direction) the graph of y = sin increases until it reaches a maximum at (90°, 1), and then
begins to decrease. Also, if decreases from 0° to –90° (in a clockwise direction), the graph
of y = sin decreases until it reaches a minium at (–90°, –1), then it begins to increase again.
The full graph of y = sin is shown below. As the shape repeats every 360°, the new sine
function is described as being periodic with period 360°.
360° corresponds to one
revolution of the circle.
y
1
y = sin
–360° –270° –180° –90° 90° 180° 270° 360°
–1
The graph of y sin extends infinitely in both directions, it
• is periodic with period 360°
• has x-intercepts at 0°, ±180°, ±360°, etc
• has turning points at = ±90°, = ±270°, etc
• is symmetric about the lines = ±90°, = ±270°, etc.
The function sin has
• natural domain R and range [–1, 1]
• a maximum value of +1 in the interval [0°, 360°] at = 90°
• a minimum value of –1 in the interval [0°, 360°] at = 270°.
7 Trigonometry II
The graph of y sin is very useful for solving trigonometric equations when there is more
than one solution.
Example
Solve the equation sin = 0.4 for 0 ≤ ≤ 360°. Find one solution using
a calculator, and the rest
Answer from the graph.
Calculator: sin = 0.4 => = 23.578°
Graph: There are two solutions between 0° and 360°.
0.4 This can be
checked on
23.578° your calculator
180°
This
By symmetry, the second solution is 180° – 23.578° = 156.422°.
Leave rounding off
The solutions are 23.6° and 156.4°. until the final answers.
Problems 2.3A
1. Solve the equation sin = 0.6 for
(a) 0 ≤ ≤ 360° (b) –360° ≤ ≤ 0°
2. Solve the equation sin = – 0.5 for
(a) –180° ≤ ≤ 180° (b) 0° ≤ ≤ 720°
The unit circle can be used to draw the graph of y cos , using the idea that the length of the
line OQ is equal to cos. To draw this graph, we need to turn the unit circle on its side.
x y
P Q
1
y O
– 90° 90°
negative positive
Sine, Cosine and Tangent 8
In the diagram above, you can see that as increases from 0° to 90° (in an anticlockwise
direction) the graph of y = cos decreases from (0°, 1) until it reaches (90°, 0), and then
continues to decrease. Also, if decreases from 0° to –90° (in a clockwise direction), the
graph of y = cos decreases from (0°, 1) until it reaches (–90°, 0), then continues to decrease.
The full graph of y = cos is shown below. It has the same shape as the graph of y =
sin translated to the left by 90°. As the shape repeats every 360°, the new cosine
function is described as being periodic with period 360°.
y
1
y = cos
–360° –270° –180° –90° 90° 180° 270° 360°
–1
The graph of y = cos extends infinitely in both directions, it
• is periodic with period 360°
• has x-intercepts at ±90°, ±270°, etc
• has turning points at = 0°, = ±180°, etc
• is symmetric about the lines = 0°, = ±180°, = ±360°, etc.
• is the translation of the graph of y = sin to the left by 90°.
The function cos has
• natural domain R and range [–1, 1]
• a maximum value of +1 in the interval [0°, 360°] at = 0° and 360°.
• a minimum value of –1 in the interval [0°, 360°] at = 180°.
9 Trigonometry II
Example
Solve the equation cos = –0.2 for 0 ≤ ≤ 360°. Find one solution using
a calculator, and the rest
Answer from the graph.
Calculator: cos = –0.2 => = 101.537°
Graph: There are two solutions between 0° and 360°.
101.537°
–0.2 360°
By symmetry, the second solution is 360° – 101.537° = 258.463°.
The solutions are –101.5° and 258.5°. Leave rounding off
until the final answers.
Problems 2.3B
1. Solve the equation cos = 0.6 for
(a) 0 ≤ ≤ 360° (b) –360° ≤ ≤ 0°
2. Solve the equation cos = – 0.5 for
(a) –180° ≤ ≤ 180° (b) 0° ≤ ≤ 720°
2.4 The Tan Function and its Graph
sin
In Module 5, we saw that tan for any angle in a right-angled triangle. We can use
cos
this relationship to define tan for any angle , provided that cos ≠ 0 ie. ≠ ±90°, etc.
Example
What are the exact value of tan 135°?
Answer
1 1
The first example in section 2.2 shows sin135 and cos135 .
2 2
Sine, Cosine and Tangent 10
sin135
so tan135
cos135
1
2
1
2
1
Problems 2.4A
Find the exact values of (a) tan 225° (b) tan 120° (b) tan(–30°)
The graph of y = tan is shown below. You can see that the shape repeats every 180°, so tan
function is periodic with period 180°.
y
4 y = tan
–360° –270° –180° –90° 90° 180° 270° 360°
–4
Notice that the tan graph has asymptotes at ±90°, ± 270°, etc. This is because of the cos in
sin
the denominator of tan . When cos is small, the value of tan is large.
cos
Example
sin 89
When = 89°, sin 89° = 0.9998 and cos 89° = 0.01745, so tan89 57.23 .
cos 89
The graph of y tan extends infinitely in both directions, it
• is periodic with period 180°
• has x-intercepts at 0°, ±180°, ±360°, etc
The function tan has natural domain R and range R.
11 Trigonometry II
Problems 2.4B
1. Solve the equation tan = 0.6 for
(a) 0 ≤ ≤ 360° (b) –360° ≤ ≤ 0°
2. Solve the equation tan = – 0.5 for
(a) –180° ≤ ≤ 180° (b) 0° ≤ ≤ 720°
3
Applications
3.1 All Stops To City
The new definitions of the sin, cos and tan functions were based upon a new way of thinking
about angles, where positive angles are measured in an anticlockwise direction and negative
angles are measured in a clockwise direction. We need to relate this the previous definitions
of sin, cos and tan ratios based on right-angled triangles.
The sin function is positive between 0° and 180°, whereas the cos function is positive
between 0° and 90°, and negative between 90° and 180°. The following diagram summarises
this information for all angles:
S A
T C
The letters A, S, T and C are interpreted as:
A All of sin, cos and tan are positive in the 1st quadrant of the coordinate
plane.
S Sin is positive in the 2nd quadrant; cos and tan are negative.
T Tan is positive in the 3rd quadrant; sin and cos are negative.
C Cos is positive in the 4th quadrant; sin and tan are negative.
The two angles in a right-angled triangle (other than the right angle) are between 0° and 90°.
As the new definitions of sin, cos and tan are all positive for angles in the first quadrant, there
is no conflict with the previous definitions of the trigonometric ratios.
12
13 Trigonometry II
3.1 Trigonometry in Triangles Without Right-angles
The sine rule is very useful in solving problems in triangles:
sin A sin B sinC A
a b c
b
c
C
B a
Example
Find the angle in the triangle below.
23°
4
2
Answer
By the sine rule,
sin sin 23
4 2
Angles between 0° and
sin 23
sin 4 90° are commonly called
2 ‘acute’, and angles
0.7815 between 90° and 180° are
called obtuse.
51.398
A second solution is 180° – 51.398° = 128.602°
The angle in the diagram is an acute angle, so = 51.4°.
The cosine rule is also very useful:
A
a b c 2bc cos A
2 2 2
b
c
C
B a
Applications 14
The cosine rule is a generalisation of Pythagoras’ Theorem. You can choose any side to have
length ‘a’, not just the longest side.
Example
Find x in the triangle below.
4 23°
2
x
Answer
By the cosine rule,
x 2 2 2 4 2 2 4 cos 23
12.636
x 3.6
Problems 3.2
Find the unknown side or angle in the triangles below.
(a) 60° (b) 60°
43°
a a
2.1 2.1
43°
(c)
4 a
40°
6
4
Identities
A trigonometric identity is a relationship between the trigonometric functions which is true
for all angles.
4.1 (sin )2 + (cos )2 = 1
This is the most famous of all trigonometric identities. It is true because the equation of the
unit circle is x2 + y2= 1.
• P (cos, sin)
1
O
sin
4.2 tan , when cos 0
cos
This is the definition of tan.
4.3 sin( ) sin
This is because the graph of y = sin is symmetric across the origin.
15
Applications 16
4.4 cos( ) cos
This is because the graph of y = cos is symmetric about the y-axis.
4.5 cos(90 ) sin; sin(90 ) cos
These identities come from the right-angled triangle.
1 90 –
sin or cos(90 –)
cos or sin(90 –)
4.6 cos( 90) sin ; sin( 90) cos
The first identity comes from the fact that translating the graph of y = cos to the right by 90°
gives the graph of y = sin
Problems 4
Use the graphs of y = sin and y = cos to explain the identity sin( 90) cos .
5
Radian Measure
The use of degrees to measure angles has its source in the astronomy of ancient times – a degree
being approximately the angle moved in one day by the earth in its journey around the sun. This is
not the best unit of measurement in mathematics, and a more convenient unit is needed for subjects
like Calculus.
The most natural unit for measuring angles is the radian. One radian is the angle in a unit circle
which subtends an arc of length 1 unit.
y
1
1c
x
The arc length around the whole unit circle is equal to its circumference ie. 2 1 units, so the
angle in a whole revolution is 2 radians (pronounced “2 pie radians”). This tells us that 2 radians
is equal to 360°, and that
radians = 180°
In mathematics we always assume that if no unit of measurement is mentioned, then the size of an
angle is in radians. Hence an angle in degrees must always have a degree symbol. If a symbol to
mean radians is necessary, many use the symbol c to mean “radians” (the c referring to
circumference).
Example
Express an angle of 1 in terms of degrees.
Answer
180
180
1
57.3
17
Radian Measure 18
Example
Express an angle of in terms of degrees.
12
Answer
180
180
12 12
15
Example
Express 15° in terms of radians.
Answer
180
1
180
15 15
180 12
Example
Express 37.9° in terms of radians.
It is a good practice to write answers
Answer as multiples of radians. This helps
180 you understand the actual size of the
angle.
1
180 In the example to the right, 0.21 can
be quickly seen to be about a fifth of
37.9 37.9
180 180°, whereas the answer 0.66 is
0.21 or 0.66 much harder to visualise.
Problems 5
1. Express the following angles in degrees.
2 3
(a) (b) 2 (c) – 2 (d) (e) (f) –
2 3 8
2. Express the following angles in radians.
30° (b) 45°
(a) (c) 60° (d) –150° (e) 540° (f) 26.5°
A
Answers
Section 1.1 Section 2.3A
(a) 1(a) 36.87°, 143.13° (b) –216.87°, –323.13°
2(a) –150°, –30° (b) 210°, 330°, 570°, 690°
(b) Section 2.3B
1(a) 53.13°, 306.87° (b) –306.87°, –53.13°
(c) 2(a) –120°, 120° (b) 120°, 240°, 480°, 600°
Section 2.4A
1
(d) 1(a) 1 (b) 3 (c)
3
Section 2.4B
(e) 1(a) 30.96°, 210.96° (b) –329.04°, -149.04°
153.43° (b) 153.43°, 333.43°,
2(a) –26.57°,
(f) 513.43°, 693.43°
Section 3.2
(g) (a) 1.7 (b) 2.4 (c) 65.4°, 6.5
Section 5
(h) 1(a) 180° (b) 360° (c) –360° (d) 90° (e) 120°
(f) –67.5°
(i)
2(a) (b) (c)
6 4 3
Section 2.2 5
(d) (e) 3 (f) 0.15
3 1 1 3 1 3 6
(a) , (b) , (c) ,
2 2 2 2 2 2
1 3 1 3 1 3
(d) , (e) , (f) ,
2 2 2 2 2 2
