Transcript Book 5 Chapter 14 Trigonometry (1)

14 Trigonometry (1)
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
Case Study
How can we find the shape of the
sound wave generated by a tuning
fork?
The sound wave generated can
be displayed by using a CRO.
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.
P. 2
14.1 Introduction to Trigonometry
A. Angles of Rotation
In the figure, the centre of the circle is O and its radius is r.
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.
P. 3
14.1 Introduction to Trigonometry
A. Angles of Rotation
Remarks:
1. 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.
P. 4
14.1 Introduction to Trigonometry
B. Quadrants
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  180
Quadrant III: 180  q  270
Quadrant IV: 270  q  360
Notes:
0, 90, 180 and 270 do not belong to any quadrant.
P. 5
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
sin q 
opposite side

hypotenuse
tan q 
y
, cos q 
r
opposite side
adjacent side

y
adjacent side
hypotenuse

x
and
r
.
x
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 have r 
2
2
x  y .
We can then define the trigonometric ratios of q in terms of x, y and r:
sin q 
y
r
, cos q 
x
r
and tan q 
y
x
P. 6
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. r  (  3 ) 2  4 2  5 .
By definition:
sin q 
y
r

4
5
cos q 
x
r
 
3
5
tan q 
y
x
 
4
3
P. 7
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.
Quadrant
I
II
III
IV
Sign of
Sign of
x-coordinate y-coordinate Sign of sin q Sign of cos q Sign of tan q




















P. 8
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’.
Quadrant
I
II
III
IV
Sign of
Sign of
x-coordinate y-coordinate Sign of sin q Sign of cos q Sign of tan q




















P. 9
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)
(b)
(c)
(d)
sin 160  0.342 (cor. to 3 sig. fig.)
tan 245  2.14 (cor. to 3 sig. fig.)
cos(123)  0.545 (cor. to 3 sig. fig.)
sin(246)  0.914 (cor. to 3 sig. fig.)
P. 10
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.

sin 90  
y

r
cos 90  
x
y
x
1
r

r
tan 90  
r
0
0
r

r
, which is undefined.
0
P. 11
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.
q
Coordinates of P
sin q
cos q
tan q
0
(r, 0)
0
1
0
90
(0, r)
1
0
undefined
180
(r, 0)
0
1
0
270
(0, r)
1
0
undefined
360
(r, 0)
0
1
0
Notes:
The terminal sides OP of q  0 and 360 lie in the same position.
Thus, their trigonometric ratios must be the same.
P. 12
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
P. 13
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.
P. 14
According to the
ASTC diagram
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
 tan 60  
3
cos 60  
1
2
Step 4: Find the trigonometric ratio of the angle 240:
 tan 240  tan 60
cos 240  cos 60

3
 
1
2 P. 15
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
sin q 
y
, cos q 
r
x
and tan q 
r
y
,
x
where P(x, y) is a point on the terminal side of the angle
of rotation q and r 
2
2
x  y 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.
P. 16
14.3 Finding Trigonometric Ratios
Without Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
Example 14.1T
If tan q  
5
12
, where 270  q  360, find the values of
sin q and cos q.
Solution:
Since tan q  0, q lies in quadrant II or IV.
As it is given that 270  q  360, q must lie in quadrant IV where
sin q  0 and cos q  0.
P(12, 5) is a point on the terminal side of q.

x  12 , y   5
By definition,

2
sin q 
r 

x  y
12
 13
2
2
 ( 5)
y
cos q 
r
2
 
5

x
r
12
13
13
P. 17
14.3 Finding Trigonometric Ratios
Without Using a Calculator
C. Finding Trigonometric Ratios by Another
Given Trigonometric Ratio
Example 14.2T
2
If sin q   , where 180  q  270, find the values of
5
cos q and tan q.
Solution:
Since sin q  0 and 180  q  270, q lies in quadrant III.
Let P(x, 2) be a point on the terminal side of q.
We have y  2 and r  5.

2
2
2
2
 21
x  (2)  5
x
x   21 or

cos q 
x

y
x
Since q lies in quadrant III, the
x-coordinate of P must be negative.
21
5
r
tan q 
21 (rejected)

2
 21

2
21
P. 18
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 quadrant II, we have
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
P. 19
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) 
1
tan q
P. 20
14.4 Trigonometric Identities
Example 14.3T
Simplify the following expressions.
(a) tan (180  q) sin (90  q)
(b)
cos (180   q )
 cos q
Solution:
(a) tan (180  q) sin (90  q)  tan q cos q

sin q
cos q
(cos q )
 sin q
(b)
cos (180   q )
 cos q

 cos q
 cos q
1
P. 21
14.4 Trigonometric Identities
Example 14.4T
Simplify sin (90  q) cos (90  q)  2sin (180  q) cos q.
Solution:
sin( 90   q ) cos( 90   q )  2 sin( 180   q ) cos q
 sin( 180   ( 90   q )) cos( 180   ( 90   q ))  2 sin q cos q
 sin( 90   q )[  cos( 90   q )]  2 sin q cos q
 cos q (  sin q )  2 sin q cos q
  sin q cos q  2 sin q cos q
  3 sin q cos q
P. 22
14.4 Trigonometric Identities
Example 14.5T
1
 sin( 270   q )
Prove that cos q 2
tan (180   q )
Solution:
L.H.S.
1
 sin( 270   q )
 cos q .
 cos q 2
tan (180   q )
1
 sin( 180   ( 90   q ))
 cos q
2
tan q
1
 sin( 90   q )
 cos q
2
tan q
1
 cos q
 cos q
2
tan q
2
1  cos q

cos q
sin
2
q
cos q
2
sin q
2

cos q
cos q
2

sin q
2
 cos q
 R.H.S.
P. 23
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 sin q  
3
2
, where 0  q  360.
Step 1: Since sin q  0, q may lie in either quadrant III or quadrant IV.
Step 2: Let b be the reference angle of q.
sin b 
3
2
 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.

q  240  or 300 
P. 24
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.
Finding the trigonometric ratio
q  120
q  120, 240, …
cos q  
1
2
cos q  
1
2
Finding the corresponding angles
P. 25
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.
P. 26
14.5 Trigonometric Equations
B. Simple Trigonometric Equations
Example 14.6T
If ( 2  1)sin q  2, where 0  q  360, find q. (Give the
answers correct to 1 decimal place.)
Solution:
( 2  1) sin q  2
sin q 
2
2 1
Hence, q  55.938 or 180  55.938
q  55 . 9  or 124.1  (cor. to 1 d. p.)
By using a calculator, the
reference angle  55.938.
P. 27
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
2 sin q  2sin q cos
q0
2cos2 q  3sin q  0
Taking out the common factor
Transforming into a quadratic equation
P. 28
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
(b)
1
cos q
2
4
Solution:
(a) 7 sin q  7 cos q  0
(b)
7 sin q   7 cos q
sin q
cos q
 
7
4
cos q
2
1  4 cos q
2
cos q 
2
7
tan q   1

1
q  135  or 315 
1
4
cos q 
1
2

or 
1
2
q  60  , 120  , 240  or 300 
P. 29
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:
cos q tan q  cos q  0
2
2  sin q 
cos q 
  cos q  0
 cos q 
cos q sin q  cos q  0
cos q (sin q  1)  0

cos q  0
q  90  or 270 

q  90  or 270 
or sin q   1
q  270 
Factorize the given
expression and apply the
fact that if ab  0, then
a  0 or b  0.
P. 30
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:
2 cos q  sin q  1  0
2
2 (1  sin
2
2  2 sin
q )  sin q  1  0
2
q  sin q  1  0
Transform the equation into
a quadratic equation with
sin q as the unknown.
 2 sin q  sin q  1  0
2
2 sin
2
q  sin q  1  0
(sin q  1)( 2 sin q  1)  0

sin q  1
or sin q  
q  90 

1
2
q  180   30  or 360   30 
q  90  , 210  or 330 
P. 31
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.
P. 32
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.
P. 33
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
y
1
0.87
0.5
90 120 150 180 210 240 270 300 330 360
0
0.5 0.87 1 0.87 0.5
0
0.5
From the above table, we can
plot the points on the coordinate
plane.
P. 34
0.87
1
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.
P. 35
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
y
0
0.58
1
x
y
60
75
0.58
1
105 120 135 150
1.73 3.73 Undefined 3.73 1.37 1 0.58
180 210 225 240 255
0
90
270
285 300 315 330 360
1.73 3.73 Undefined 3.73 1.37 1 0.58
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.
P. 36
0
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
y
0
0.58
1
x
y
60
75
0.58
1
105 120 135 150
1.73 3.73 Undefined 3.73 1.37 1 0.58
180 210 225 240 255
0
90
270
285 300 315 330 360
1.73 3.73 Undefined 3.73 1.37 1 0.58
P. 37
0
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, ... .
P. 38
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.
P. 39
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.
P. 40
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)
y  cos x
y  cos (x  180)
(iii) y  cos x
Solution:
y  cos x  2
(a) Refer to the figure.
(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.
P. 41
14.7 Graphical Solutions of
Trigonometric Equations
Similar to quadratic equations, trigonometric equations can be
solved either by the algebraic method or the graphical method.
We should note that the graphical solutions are approximate in nature.
P. 42
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.
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.
P. 43
y  0.7
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
y  1.5
Solution:
(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.
P. 44
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.
P. 45
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.
sin q 
cos q 
tan q 
y
r
x
r
y
x
P. 46
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.
P. 47
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
P. 48
Chapter Summary
14.5 Trigonometric Equations
Trigonometric equations can be solved by the algebraic method.
P. 49
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.
The periods of sin x, cos x and tan x are 360, 360 and 180
respectively.
5.
P. 50
Chapter Summary
14.7 Graphical Solutions of Trigonometric
Equations
Trigonometric equations can be solved by the graphical method.
P. 51