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Title: Trigonometry (1)

1
Trigonometry (1)
14
Case Study
14.1 Introduction to Trigonometry
14.2 Trigonometry Ratios of Arbitrary Angles
14.3 Finding Trigonometric Ratios Without
Using a Calculator
14.4 Trigonometric Identities
14.5 Trigonometric Equations
14.6 Graphs of Trigonometric Functions
14.7 Graphical Solutions of Trigonometric
Equations
Chapter Summary
2
Case Study
The figure shows the sound wave generated by the
tuning fork displayed on a cathode-ray
oscilloscope (CRO).
The pattern of the waveform of sound has the same
shape as the graph of a trigonometric function.
The graph repeats itself at regular intervals.
Such an interval is called the period.
3
14.1 Introduction to Trigonometry
A. Angles of Rotation
In the figure, the centre of the circle is O and
Suppose OA is rotated about O and it reaches OP,
the angle q formed is called an angle of
rotation.
• OA initial side
• OP terminal side

If OA is rotated in an anti-clockwise direction,
the value of q is positive.
If OA is rotated in a clockwise direction, then
the value of q is negative.
4
14.1 Introduction to Trigonometry
A. Angles of Rotation
• Remarks
• The figure shows the measures of two different
angles 130? and ?230?.
• However, they have the same initial side OA and
terminal side OP.

2. The initial side and terminal side of 410?
coincide with that of 50? as shown in the figure.
5
14.1 Introduction to Trigonometry
In a rectangular coordinate plane, the x-axis and
the y-axis divide the plane into four parts as
shown in the figure.
Each part is called a quadrant.
Notes The x-axis and the y-axis do not belong
to any of the four quadrants.
For an angle of rotation, the position where the
terminal side lies determines the quadrant in
which the angle lies.
Thus, we can see that for an angle of rotation q,
Quadrant I 0? ? q ? 90? Quadrant II 90? ? q ?
IV 270? ? q ? 360?
Notes 0?, 90?, 180? and 270? do not belong to
6
14.2 Trigonometric Ratios of Arbitrary
Angles
A. Definition
For an acute angle q, the trigonometric ratios
between two sides of a right-angled triangle are
We now introduce a rectangular coordinate plane
onto DOPQ such that OP is the terminal side as
shown in the figure.
Suppose the coordinates of P are (x, y) and the
length of OP is r.
We can then define the trigonometric ratios of q
in terms of x, y and r
7
14.2 Trigonometric Ratios of Arbitrary
Angles
A. Definition
Now, we can extend the definition for angles
greater than 90?.
For example In the figure, P(3 , 4) is a point
on the terminal side of the angle of rotation q.
We have x ? ?3 and y ? 4.
By definition
8
14.2 Trigonometric Ratios of Arbitrary
Angles
B. Signs of Trigonometric Ratios
In the previous section, we defined the
trigonometric ratios in terms of the coordinates
of a point P(x, y) on the terminal side and the
length r of OP.
Since x and y may be either positive or negative,
the trigonometric ratios may be either positive
or negative depending upon the quadrant in which
q lies.
?
?
?
?
?
I
?
?
?
?
?
II
?
?
?
?
?
III
?
?
?
?
?
IV
9
14.2 Trigonometric Ratios of Arbitrary
Angles
B. Signs of Trigonometric Ratios
The signs of the three trigonometric ratios in
different quadrants can be summarized in the
following diagram which is called an ASTC diagram.
A All positive S Sine positive T Tangent
positive C Cosine positive
Notes ASTC can be memorized as Add Sugar To
Coffee.
10
14.2 Trigonometric Ratios of Arbitrary
Angles
C. Using a Calculator to Find Trigonometric
Ratios
We can find the trigonometric ratios of given
angles by using a calculator.
For example,
(a) sin 160? ? 0.342 (cor. to 3 sig. fig.)
(b) tan 245? ? 2.14 (cor. to 3 sig. fig.)
(c) cos(?123?) ? ?0.545 (cor. to 3 sig. fig.)
(d) sin(?246?) ? 0.914 (cor. to 3 sig. fig.)
11
14.3 Finding Trigonometric Ratios Without
Using a Calculator
A. Angles Formed by Coordinates Axes
If we rotate the terminal side OP with length r
units (r ? 0) through 90? in an anti-clockwise
direction, then the coordinates of P are (0, r).
Thus, x ? 0 and y ? r.
, which is undefined.
12
14.3 Finding Trigonometric Ratios Without
Using a Calculator
A. Angles Formed by Coordinates Axes
Suppose we rotate the terminal side OP through
90?, 180?, 270? and 360? in an anti-clockwise
direction.
0
?1
0
(?r, 0)
180?
undefined
0
?1
(0, ?r)
270?
0
1
0
(r, 0)
360?
Notes The terminal sides OP of q ? 0? and
360? lie in the same position. Thus, their
trigonometric ratios must be the same.
13
14.3 Finding Trigonometric Ratios Without
Using a Calculator
B. By Considering the Reference Angles
1. Reference Angle
For each angle of rotation q (except for q ? 90?
? n, where n is an integer), we consider the
corresponding acute angle measured between the
terminal side and the x-axis.
It is called the reference angle b.
Examples
? q ? 30? ? b ? 30?
? q ? 140? ? b ? 180? ? 140? ? 40?
? q ? 250? ? b ? 250? ? 180? ? 70?
? q ? 310? ? b ? 360? ? 310? ? 50?
14
14.3 Finding Trigonometric Ratios Without
Using a Calculator
B. By Considering the Reference Angles
2. Finding Trigonometric Ratios
By using the reference angle, we can find the
trigonometric ratios of an arbitrary angle.
The following four steps can help us find the
trigonometric ratio of any given angle q
Step 1 Determine the quadrant in which the angle
q lies.
Step 2 Determine the sign of the corresponding
trigonometric ratio.
Step 3 Find the trigonometric ratio of its
reference angle b.
Step 4 Find the trigonometric ratio of the angle
q by assigning the sign determined in step 2 to
the ratio determined in step 3.
15
14.3 Finding Trigonometric Ratios Without
Using a Calculator
B. By Considering the Reference Angles
For example, to find tan 240? and cos 240?
Step 1 Determine the quadrant in which the angle
240? lies
? 240? lies in quadrant III.
Step 2 Determine the sign of the corresponding
trigonometric ratio
? In quadrant III tangent ratio ?ve
cosine ratio ?ve
\ tan q ? tan b cos q ? ?cos b
Step 3 Find the trigonometric ratio of its
reference angle b
? b ? 240? ? 180? ? 60?
Step 4 Find the trigonometric ratio of the angle
240?
? tan 240? ? tan 60? cos 240? ? ?cos 60?
16
14.3 Finding Trigonometric Ratios Without
Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
In the last section, we learnt that the
trigonometric ratios can be defined as
where P(x, y) is a point on the terminal side of
the angle of rotation q and
is the length of OP.
Now, we can use the above definitions to find
other trigonometric ratios of an angle when one
of the trigonometric ratios is given.
17
14.3 Finding Trigonometric Ratios Without
Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
Example 14.1T
If , where 270? ? q ? 360?,
find the values of sin q and cos q.
Solution
Since tan ? ? 0, ? lies in quadrant II or IV.
As it is given that 270? ? ? ? 360?, ? must lie
in quadrant IV where sin ? ? 0 and cos ? ? 0.
P(12, ?5) is a point on the terminal side of ?.
By definition,
18
14.3 Finding Trigonometric Ratios Without
Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
Example 14.2T
If , where 180? ? q ? 270?,
find the values of cos q and tan q.
Solution
Since sin ? ? 0 and 180? ? ? ? 270?, ? lies in
Let P(x, ?2) be a point on the terminal side of ?.
We have y ? ?2 and r ? 5.
19
14.4 Trigonometric Identities
With the help of reference angles in the last
section, we can get the following important
identities.
For any acute angle q, since 180? ? q lies in
sin (180? ? q) ? sin q cos (180? ? q) ? ?cos q
tan (180? ? q) ? ?tan q
Since 180? ? q lies in quadrant III, we have
sin (180? ? q) ? ?sin q cos (180? ? q) ? ?cos q
tan (180? ? q) ? tan q
20
14.4 Trigonometric Identities
Since 360? ? q lies in quadrant IV, we have
sin (360? ? q) ? ?sin q cos (360? ? q) ? cos q
tan (360? ? q) ? ?tan q
Notes The above identities also hold if q is
not an acute angle.
They are useful in simplifying expressions
involving trigonometric ratios.
Remarks The following identities also hold if q
is not an acute angle sin (90? ? q) ? cos q
cos (90? ? q) ? sin q tan (90? ? q) ?
21
14.4 Trigonometric Identities
Example 14.3T
Simplify the following expressions. (a) tan
(180? ? q) sin (90? ? q)
Solution
(a) tan (180? ? q) sin (90? ? q)
22
14.4 Trigonometric Identities
Example 14.4T
Simplify sin (90? ? q) cos (90? ? q) ? 2sin (180?
? q) cos q.
Solution
23
14.4 Trigonometric Identities
Example 14.5T
Solution
24
14.5 Trigonometric Equations
A. Finding Angles from Given Trigonometric
Ratios
In previous sections, we learnt how to find the
trigonometric ratios of any angle.
Now, we will study how to find the angle if a
trigonometric ratio of the angle is given. For
example
Given that , where 0? ? q ?
360?.
Step 1 Since sin q ? 0, q may lie in either
Step 2 Let b be the reference angle of q.
? b ? 60?
Step 3 Locate the angle q and its reference
angle b in each possible quadrant.
Step 4 Hence, if q lies in quadrant III,
q ? 180? ? 60? ? 240?.
If q lies in quadrant IV,
q ? 360? ? 60? ? 300?.
25
14.5 Trigonometric Equations
A. Finding Angles from Given Trigonometric
Ratios
In general, for any given trigonometric ratio, it
may correspond to more than one angle.
26
14.5 Trigonometric Equations
B. Simple Trigonometric Equations
An equation involving trigonometric ratios of an
unknown angle q is called a trigonometric
equation.
Usually, there are certain values of q which
satisfy the given equation.
The process of finding the solutions of the
equation is called solving trigonometric
equation.
We will try to solve some simple trigonometric
equations a sin q ? b, a cos q ? b and a tan q ?
b, where a and b are real numbers.
27
14.5 Trigonometric Equations
B. Simple Trigonometric Equations
Example 14.6T
If ( ? 1)sin q ? 2, where 0? ? q ? 360?,
find q. (Give the answers correct to 1 decimal
place.)
Solution
Hence, ? ? 55.938? or 180? ? 55.938?
(cor. to 1 d. p.)
28
14.5 Trigonometric Equations
C. Other Trigonometric Equations
We now try to solve some harder trigonometric
equations.
Examples
Equation Technique
2sin q ? 3cos q ? 0 Using trigonometric identity
5sin2 q ? 4 ? 0 Taking square root
sin q ? 2sin q cos q ? 0 Taking out the common factor
2cos2 q ? 3sin q ? 0 Transforming into a quadratic equation
29
14.5 Trigonometric Equations
C. Other Trigonometric Equations
Example 14.7T
Solve the following equations for 0? ? q ? 360?.
(a) 7sin q ? 7cos q ? 0
Solution
30
14.5 Trigonometric Equations
C. Other Trigonometric Equations
Example 14.8T
Solve the equation cos2 q tan q ? cos q ? 0 for
0? ? q ? 360?.
Solution
31
14.5 Trigonometric Equations
C. Other Trigonometric Equations
Example 14.9T
Solve the equation 2cos2 q ? sin q ? 1 ? 0 for 0?
? q ? 360?.
Solution
32
14.6 Graphs of Trigonometric Functions
A. The Graph of y ? sin x
Consider y ? sin x. For every angle x, there is a
corresponding trigonometric ratio y. Thus, y is a
function of x.
The following table shows some values of x and
the corresponding values of y (correct to 2
decimal places if necessary) for 0? x 360?.
x 0? 30? 60? 90? 120? 150? 180?
y 0 0.5 0.87 1 0.87 0.5 0
x 210? 240? 270? 300? 330? 360?
y ?0.5 ?0.87 ?1 ?0.87 ?0.5 0
From the above table, we can plot the points on
the coordinate plane.
33
14.6 Graphs of Trigonometric Functions
A. The Graph of y ? sin x
We can also plot the graph of y ? sin x for 360?
x 720?, etc.
The graph of y ? sin x repeats itself in the
intervals 360? x 0?, 0? x 360?, 360? x
720?, etc.
Remarks A function repeats itself at regular
intervals is called a periodic function.
The regular interval is called a period.
From the figure, we obtain the following results
for the graph of y ? sin x for 0? x 360?
1. The domain of y ? sin x is the set of all real
numbers.
2. The maximum value of y is 1, which corresponds
to x ? 90?. The minimum value of y is 1, which
corresponds to x ? 270?.
3. The function is a periodic function with a
period of 360?.
34
14.6 Graphs of Trigonometric Functions
B. The Graph of y ? cos x
The following table shows some values of x and
the corresponding values of y (correct to 2
decimal places if necessary) for 0? x 360?
for y ? cos x.
x 0? 30? 60? 90? 120? 150? 180? 210? 240? 270? 300? 330? 360?
y 1 0.87 0.5 0 ?0.5 ?0.87 ?1 ?0.87 ?0.5 0 0.5 0.87 1
From the above table, we can plot the points on
the coordinate plane.
35
14.6 Graphs of Trigonometric Functions
B. The Graph of y ? cos x
From the figure, we obtain the following results
for the graph of y ? cos x for 0? x 360?
1. The domain of y ? cos x is the set of all real
numbers.
2. The maximum value of y is 1, which corresponds
to x ? 0? and 360?. The minimum value of y is
1, which corresponds to x ? 180?.
Notes If we plot the graph of y ? cos x for
360? x 720?, we can see that the graph
repeats itself every 360?. Thus, y ? cos x is a
periodic function with a period of 360?.
36
14.6 Graphs of Trigonometric Functions
C. The Graph of y ? tan x
The following table shows some values of x and
the corresponding values of y (correct to 2
decimal places if necessary) for 0? x 360?
for y ? tan x.
x 0? 30? 45? 60? 75? 90? 105? 120? 135? 150?
y 0 0.58 1 1.73 3.73 Undefined ?3.73 ?1.37 ?1 ?0.58
x 180? 210? 225? 240? 255? 270? 285? 300? 315? 330? 360?
y 0 0.58 1 1.73 3.73 Undefined ?3.73 ?1.37 ?1 ?0.58 0
The value of y is not defined when x ? 90? and
270?.
When an angle is getting closer and closer to 90?
or 270?, the corresponding value of tangent
function approaches to either positive infinity
or negative infinity.
37
14.6 Graphs of Trigonometric Functions
C. The Graph of y ? tan x
The graph of y ? tan x is drawn as below.
x 0? 30? 45? 60? 75? 90? 105? 120? 135? 150?
y 0 0.58 1 1.73 3.73 Undefined ?3.73 ?1.37 ?1 ?0.58
x 180? 210? 225? 240? 255? 270? 285? 300? 315? 330? 360?
y 0 0.58 1 1.73 3.73 Undefined ?3.73 ?1.37 ?1 ?0.58 0
38
14.6 Graphs of Trigonometric Functions
C. The Graph of y ? tan x
From the figure, we obtain the following results
for the graph of y ? tan x
1. For 0? x 180?, y ? tan x exhibits the
following behaviours
From 0? to 90?, tan x increases from 0 to
positive infinity. From 90? to 180?, tan x
increases from negative infinity to 0.
2. y ? tan x is a periodic function with a period
of 180?.
3. As tan x is undefined when x ? 90? and 270?,
the domain of y ? tan x is the set of all real
numbers except x ? 90?, 270?, ... .
39
14.6 Graphs of Trigonometric Functions
C. The Graph of y ? tan x
Given a trigonometric function, we can find its
maximum and minimum values algebraically.
For example, to find the maximum and minimum
values of 3 ? 4cos x
?1 ? cos x ? 1
?4 ? 4cos x ? 4
?4 ? 3 ? 3 ? 4cos x ? 4 ? 3
?1 ? 3 ? 4cos x ? 7
The maximum and minimum values are 7 and ?1
respectively.
40
14.6 Graphs of Trigonometric Functions
D. Transformation on the Graphs of
Trigonometric Functions
In Book 4, we learnt the transformations such as
translation and reflection of graphs of
functions.
Now, we will study the transformations on the
graphs of trigonometric functions.
41
14.6 Graphs of Trigonometric Functions
D. Transformation on the Graphs of
Trigonometric Functions
Example 14.10T
(a) Sketch the graph of y ? cos x for ?180? x
360?. (b) From the graph in (a), sketch the
graphs of the following functions. (i) y ? cos
x ? 2 (ii) y ? cos (x ? 180?) (iii) y ? ?cos x
y ? cos (x ? 180?)
y ? ?cos x
Solution
(a) Refer to the figure.
y ? cos x ? 2
(b) The graph of the function (i) y ? cos x ? 2
is obtained by translating the graph of y ? cos
x two units downwards.
(ii) y ? cos (x ? 180?) is obtained by
translating the graph of y ? cos x to the left
by 180?.
(ii) y ? ?cos x is obtained by reflecting the
graph of y ? cos x about the x-axis.
42
14.7 Graphical Solutions of
Trigonometric Equations
equations can be solved either by the algebraic
method or the graphical method.
We should note that the graphical solutions are
approximate in nature.
43
14.7 Graphical Solutions of
Trigonometric Equations
Example 14.11T
Consider the graph of y ? cos x? for 0 x 360.
Using the graph, solve the following equations.
(a) cos x? ? 0.6 (b) cos x? ? ?0.7
y ? 0.6
Solution
(a) Draw the straight line y ? 0.6 on the graph.
The straight line cuts the curve at x ? 54 and
306.
y ? ?0.7
So the solution of cos x? ? 0.6 for 0 x 360
is 54 or 306.
(b) Draw the straight line y ? ?0.7 on the graph.
The straight line cuts the curve at x ? 135 and
225.
So the solution of cos x? ? ?0.7 for 0 x 360
is 135 or 225.
44
14.7 Graphical Solutions of
Trigonometric Equations
Example 14.12T
Draw the graph of y ? 3cos x? ? sin x? for 0 x
360. Using the graph, solve the following
equations for 0 x 360. (a) 3cos x? ? sin x?
? 0 (b) 3cos x? ? sin x? ? 1.5
Solution
y ? 1.5
(a) From the graph, the curve cuts the x-axis at
x ? 72 and 252.
Therefore, the solution is 72 or 252.
(b) Draw the straight line y ? 1.5 on the graph.
The straight line cuts the curve at x ? 43 and
280.
Therefore, the solution is 43 or 280.
45
Chapter Summary
14.1 Introduction to Trigonometry
In a rectangular coordinate plane, the x-axis and
the y-axis divide the plane into four quadrants.
46
Chapter Summary
14.2 Trigonometric Ratios of Arbitrary Angles
The signs of different trigonometric ratios in
different quadrants can be memorized by the ASTC
diagram.
47
Chapter Summary
14.3 Finding Trigonometric Ratios Without Using
a Calculator
If b is the reference angle of an angle q,
then sin q ? ?sin b, cos q ? ?cos b, tan q ?
?tan b, where the choice of the sign (? or ?)
depends on the quadrant in which q lies.
48
Chapter Summary
14.4 Trigonometric Identities
1. (a) sin (180? q) ? sin q (b) cos (180? q)
? cos q (c) tan (180? q) ? tan q
2. (a) sin (180? ? q) ? sin q (b) cos (180? ?
q) ? cos q (c) tan (180? ? q) ? tan q
3. (a) sin (360? q) ? sin q (b) cos (360?
q) ? cos q (c) tan (360? q) ? tan q
49
Chapter Summary
14.5 Trigonometric Equations
Trigonometric equations can be solved by the
algebraic method.
50
Chapter Summary
14.6 Graphs of Trigonometric Functions
1. Graph of y ? sin x
2. Graph of y ? cos x
3. Graph of y ? tan x
4. For any real value of x, ?1 ? sin x? ?
1 and ?1 ? cos x? ? 1.
5. The periods of sin x, cos x and tan x are
360?, 360? and 180? respectively.
51
Chapter Summary
14.7 Graphical Solutions of Trigonometric
Equations
Trigonometric equations can be solved by the
graphical method.
52
Follow-up 14.1
14.3 Finding Trigonometric Ratios Without
Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
If , where 180? ? q ? 270?, find
the values of sin q and cos q.
Solution
Since tan ? ? 0, ? lies in quadrant I or III.
As it is given that 180? ? ? ? 270?, ? must lie
in quadrant III where sin ? ? 0 and cos ? ? 0.
P(?4, ?3) is a point on the terminal side of ?.
By definition,
53
Follow-up 14.2
14.3 Finding Trigonometric Ratios Without
Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
If , where 90? ? q ? 180?,
find the values of sin q and tan q.
Solution
Since cos ? ? 0 and 90? ? ? ? 180?, ? lies in
Let P(?3, y) be a point on the terminal side of ?.
We have x ? ?3 and r ? 5.
54
Follow-up 14.3
14.4 Trigonometric Identities
Simplify the following expressions. (a) cos
(180? ? q) tan (180? ? q)
Solution
(a) cos (180? ? q) tan (180? ? q)
55
Follow-up 14.4
14.4 Trigonometric Identities
Simplify tan (90? ? q) sin (180? ? q) ? 4cos
(180? ? q).
Solution
56
Follow-up 14.5
14.4 Trigonometric Identities
Solution
57
Follow-up 14.6
14.5 Trigonometric Equations
B. Simple Trigonometric Equations
Solve 5cos q ? ?2, where 0? ? q ? 360?. (Give the
answers correct to 1 decimal place.)
Solution
Hence, ? ? 180? ? 66.422? or 180? ? 66.422?
(cor. to 1 d. p.)
58
Follow-up 14.7
14.5 Trigonometric Equations
C. Other Trigonometric Equations
Solve the following equations for 0? ? q ?
360?. (Give the answers correct to 1 decimal
place.) (a) 6sin q ? 8cos q ? 0 (b) tan2 q ? 2 ? 0
Solution
(cor. to 1 d. p.)
(cor. to 1 d. p.)
59
Follow-up 14.8
14.5 Trigonometric Equations
C. Other Trigonometric Equations
Solve the equation tan q ? 2sin q ? 0 for 0? ? q
? 360?.
Solution
60
Follow-up 14.9
14.5 Trigonometric Equations
C. Other Trigonometric Equations
Solve the equation 2sin2 q ? 3cos q ? 0 for 0? ?
q ? 360?.
Solution
61
Follow-up 14.10
14.6 Graphs of Trigonometric Functions
D. Transformation on the Graphs of
Trigonometric Functions
The following figure shows the graph of the
function y ? tan x for 90? x 540?. Sketch the
graphs of the following functions on the figure.
(i) y ? tan x ? 1 (ii) y ? ?tan x
y ? ?tan x
Solution
y ? tan x ? 1
(i) The graph of the function y ? tan x ? 1 is
obtained by translating the graph of y ? tan x
one unit downwards.
(ii) The graph of the function y ? ?tan x is
obtained by reflecting the graph of y ? tan x
62
Follow-up 14.11
14.7 Graphical Solutions of
Trigonometric Equations
Consider the graph of y ? tan x? for 0 x 360.
Using the graph, solve the following equations.
(a) tan x? ? 3 (b) tan x? ? ?2
y ? 3
Solution
(a) Draw the straight line y ? 3 on the graph.
The straight line cuts the curve at x ? 72 and
252.
y ? ?2
So the solution of tan x? ? 3 for 0 x 360 is
72 or 252.
(b) Draw the straight line y ? ?2 on the graph.
The straight line cuts the curve at x ? 117 and
297.
So the solution of tan x? ? ?2 for 0 x 360
is 117 or 297.
63
Follow-up 14.12
14.7 Graphical Solutions of
Trigonometric Equations
The figure shows the graph of y ? acos x? ? bsin
x? for 0 x 360. (a) Find the values of a and
b. (b) Using the graph, solve the
equation ?2cos x? ? 3sin x? ? 2.
y ? 2
Solution
(a) Since the graph passes through (0, ?2), we
have
(b) Draw the straight line y ? 2 on the graph.
Since the graph passes through (90, 3), we have
The straight line cuts the curve at x ? 66 and
180.
Therefore, the solution is 66 or 180.