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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

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.

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.

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.

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.

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

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

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

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.

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.)

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.

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.

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?

14.3 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.

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?

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.

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,

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

quadrant III.

Let P(x, ?2) be a point on the terminal side of ?.

We have y ? ?2 and r ? 5.

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

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) ?

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)

14.4 Trigonometric Identities

Example 14.4T

Simplify sin (90? ? q) cos (90? ? q) ? 2sin (180?

? q) cos q.

Solution

14.4 Trigonometric Identities

Example 14.5T

Solution

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

quadrant III or quadrant IV.

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?.

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.

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.

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.)

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

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

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

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

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.

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?.

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.

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?.

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.

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

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?, ... .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

Chapter Summary

14.5 Trigonometric Equations

Trigonometric equations can be solved by the

algebraic method.

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.

Chapter Summary

14.7 Graphical Solutions of Trigonometric

Equations

Trigonometric equations can be solved by the

graphical method.

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,

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

quadrant II.

Let P(?3, y) be a point on the terminal side of ?.

We have x ? ?3 and r ? 5.

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)

Follow-up 14.4

14.4 Trigonometric Identities

Simplify tan (90? ? q) sin (180? ? q) ? 4cos

(180? ? q).

Solution

Follow-up 14.5

14.4 Trigonometric Identities

Solution

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.)

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.)

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

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

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

about the x-axis.

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.

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.