Trigonometry (1)

14

14.1 Introduction to Trigonometry 14.2 Trigonometry Ratios of Arbitrary Angles14.3 Finding Trigonometric Ratios Without Using

a Calculator

Chapter Summary

Case Study

Trigonometry (1)

14.4 Trigonometric Identities14.5 Trigonometric Equations14.6 Graphs of Trigonometric Functions14.7 Graphical Solutions of Trigonometric Equations

P. 2

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.

Case StudyCase Study

The graph repeats itself at regular intervals.

Such an interval is called the period.

How can we find the shape of the sound wave generated by a tuning fork?

The sound wave generated canbe displayed by using a CRO.

P. 3

In the figure, the centre of the circle is O and its radius is r.

1144 .1 .1 Introduction to TrigonometryIntroduction to Trigonometry

A. A. Angles of RotationAngles of Rotation

Suppose OA is rotated about O and it reaches OP, the angle 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 is positive.

If OA is rotated in a clockwise direction, then the value of is negative.

P. 4

Remarks: 1. The figure shows the measures of two different angles: 1

30 and 230.However, they have the same initial side OA and terminal side OP.


A. A. Angles of RotationAngles of Rotation

2. The initial side and terminal side of 410 coincide with that of 50 as shown in the figure.

P. 5

In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four parts as shown in the figure.


BB. . QuadrantsQuadrants

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 ,

Quadrant I: 0 90Quadrant II: 90 180 Quadrant III: 180

270Quadrant IV: 270

360 Notes: 0, 90, 180 and 270 do not belong to any quadrant.

P. 6

For an acute angle , the trigonometric ratios between two sides of a right-angled triangle are

1144 ..22 Trigonometric Ratios of ArbitraryTrigonometric Ratios of Arbitrary AnglesAngles

AA. . DefinitionDefinition

.sideadjacent

side oppositetan

x

y

and hypotenuse

sideadjacent cos

r

x,hypotenuse

side oppositesin

r

y

We now introduce a rectangular coordinate plane onto OPQ 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.

. have We 22 yxr We can then define the trigonometric ratios of in terms of x, y and r:

x

y

r

x

r

y tanand cos,sin

P. 7

For example:In the figure, P(–3 , 4) is a point on the terminal side of the angle of rotation .


AA. . DefinitionDefinition

We have x 3 and y 4.

.54)3( 22 r

5

4sin

r

y3

4tan

x

y5

3cos

r

x

Now, we can extend the definition for angles greater than 90.

By definition:

P. 8

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.


BB. . Signs of Trigonometric RatiosSigns of Trigonometric Ratios

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 lies.

IV

III

II

I

Sign of tanSign of cos Sign of sin Sign ofy-coordinate

Sign ofx-coordinateQuadrant

P. 9


BB. . Signs of Trigonometric RatiosSigns of Trigonometric Ratios

A : All positive S : Sine positive T : Tangent positive C : Cosine positive

Notes: ‘ASTC’ can be memorized as ‘Add Sugar To Coffee’.

IV

III

II

I

Sign of tanSign of cos Sign of sin Sign ofy-coordinate

Sign ofx-coordinateQuadrant

The signs of the three trigonometric ratios in different quadrants can be summarized in the following diagram which is called an ASTC diagram.

P. 10

We can find the trigonometric ratios of given angles by using a calculator.


CC. . Using a Calculator to Find TrigonometricUsing a Calculator to Find Trigonometric RatiosRatios

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.)

P. 11

1144 ..33 Finding Trigonometric RatiosFinding Trigonometric Ratios Without Using a CalculatorWithout Using a Calculator

AA. . Angles Formed by Coordinates AxesAngles Formed by Coordinates Axes

Thus, x 0 and y r.

090tan

r

x

y

00

90cos rr

x

190sin r

r

r

y

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).

, which is undefined.

P. 12


AA. . Angles Formed by Coordinates AxesAngles Formed by Coordinates Axes

010(r, 0)360

undefined01(0, r)270

010(r, 0)180

undefined01(0, r)90

010(r, 0)0

tan cos sin Coordinates of P

Notes: The terminal sides OP of 0 and 360 lie in the same position. Thus, their trigonometric ratios must be the same.

Suppose we rotate the terminal side OP through 90, 180, 270 and 360 in an anti-clockwise direction.

P. 13

1. Reference Angle


BB. . By Considering the Reference AnglesBy Considering the Reference Angles

For each angle of rotation (except for 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 .

Examples:

140 180 140 40

310 360 310 50

30 30

250 250 180 70

P. 14

2. Finding Trigonometric Ratios



By using the reference angle, we can find the trigonometric ratios of an arbitrary angle.

Step 1: Determine the quadrant in which the angle lies.

Step 2: Determine the sign of the corresponding trigonometric ratio.

Step 3: Find the trigonometric ratio of its reference angle .

Step 4: Find the trigonometric ratio of the angle by assigning the sign determined in step 2 to the ratio determined in step 3.

The following four steps can help us find the trigonometric ratio of any given angle :

According to theASTC diagram

P. 15



For example, to find tan 240 and cos 240:

3

Step 1: Determine the quadrant in which the angle 240 lies:

Step 2: Determine the sign of the corresponding trigonometric ratio:

240 lies in quadrant III.

In quadrant III: tangent ratio: ve

tan tan cos cos Step 3: Find the trigonometric ratio of its reference angle :

240 180 60

Step 4: Find the trigonometric ratio of the angle 240: tan 240 tan 60 cos 240 cos 60

2

1

cosine ratio: ve

360tan 2

160cos

P. 16


CC. . Finding Trigonometric Ratios by AnotherFinding Trigonometric Ratios by Another Given Trigonometric RatioGiven Trigonometric Ratio

where P(x, y) is a point on the terminal side of the angle

of rotation 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.

22 yxr

In the last section, we learnt that the trigonometric ratios can be defined as

,tanandcos,sinx

y

r

x

r

y

P. 17

Example 14.1T

Solution:


If , where 270 360, find the values of

sin and cos . 12

5tan

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.

5,12 yx

P(12, 5) is a point on the terminal side of .

22 yxr r

ysin

13

513

)5(12 22

r

xcos

13

12


By definition,

P. 18

Example 14.2T


222 5)2( x

212 x(rejected)21or 21x

r

x cos5

21

x

ytan

21

2


If , where 180 270, find the values of

cos and tan . 5

2sin

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.

Since lies in quadrant III, thex-coordinate of P must be negative.

21

2

P. 19

1144 ..44 Trigonometric IdentitiesTrigonometric Identities

For any acute angle , since 180 lies in quadrant II, we have

With the help of reference angles in the last section, we can get the following important identities.

Since 180 lies in quadrant III, we have

sin (180 ) sin cos (180 ) cos tan (180 ) tan


P. 20


Notes: The above identities also hold if is not an acute angle. They are useful in simplifying expressions involving trigonometric ratios.

Since 360 lies in quadrant IV, we have


Remarks: The following identities also hold if is not an acute angle:

sin (90 ) cos cos (90 ) sin

tan (90 ) tan

1

P. 21

Example 14.3T

Solution:


Simplify the following expressions. (a) tan (180 ) sin (90 )

cos

)180(cos(b)

sin

costan

)(coscos

sin

cos

)180(cos(b)

cos

cos

1

(a) tan (180 ) sin (90 )

P. 22

Example 14.4T


cos)180sin(2)90cos()90sin( cossin2))90(180cos())90(180sin(

cossin2)sin(cos cossin2cossin

cossin3

cossin2)]90cos()[90sin(

Simplify sin (90 ) cos (90 ) 2sin (180 ) cos .

Solution:

P. 23

Example 14.5T


.cos)180(tan

)270sin(cos

1

thatProve2

)180(tan

)270sin(cos

1

L.H.S.2

2tan

))90(180sin(cos

1

2tan

)90sin(cos

1

2tan

coscos

1

2

2

2

cos

sincos

cos1

2

22

sin

cos

cos

sin

cos

Solution:

R.H.S.

P. 24

1144 ..55 Trigonometric EquationsTrigonometric Equations

A. Finding Angles from Given TrigonometricA. Finding Angles from Given Trigonometric RatiosRatios

Given that , where 0 360. 2

3sin

Step 1: Since sin 0, may lie in either quadrant III or quadrant IV.

Step 2: Let be the reference angle of .

2

3sin 60

Step 3: Locate the angle and its reference angle in each possible quadrant.

Step 4: Hence, if lies in quadrant III, 180 60 240. If lies in quadrant IV, 360 60 300.

300or 240

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:

P. 25


A. Finding Angles from Given TrigonometricA. Finding Angles from Given Trigonometric RatiosRatios

In general, for any given trigonometric ratio, it may correspond to more than one angle.

120

120, 240, …

Finding the trigonometric ratio

2

1cos

2

1cos

Finding the corresponding angles

P. 26


B. Simple Trigonometric EquationsB. Simple Trigonometric Equations

An equation involving trigonometric ratios of an unknown angle is called a trigonometric equation.

Usually, there are certain values of 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 b, a cos b and a tan b, where a and b are real numbers.

P. 27

Example 14.6T


B. Simple Trigonometric EquationsB. Simple Trigonometric Equations

2

(cor. to 1 d. p.)

2sin)12(

12

2sin

Hence, 55.938 or 180 55.938 124.1or 9.55

If ( 1)sin 2, where 0 360, find . (Give the answers correct to 1 decimal place.)

Solution:

By using a calculator, the reference angle 55.938.

P. 28


C. Other Trigonometric EquationsC. Other Trigonometric Equations

We now try to solve some harder trigonometric equations.

Equation Technique

2sin 3cos 0 Using trigonometric identity

5sin2 4 0 Taking square root

sin 2sin cos 0 Taking out the common factor

2cos2 3sin 0 Transforming into a quadratic equation

Examples:

2

P. 29

Example 14.7T


0cos7sin7 (a) cos7sin7

7

7

cos

sin

1tan 315or 135

4cos

1(b)

2

2cos41

4

1cos2

2

1or

2

1cos

300or 240,120,60


Solve the following equations for 0 360.

(a) 7sin 7cos 0

Solution:

4cos

1 (b)

2

P. 30

Example 14.8T


0costancos2

0coscos

sincos2

0cossincos 0)1(sincos

1sinor 0cos

270or 90


Solve the equation cos2 tan cos 0 for 0 360.Solution:

Factorize the given expression and apply the fact that if ab 0, thena 0 or b 0.

270or 90 270

P. 31

Example 14.9T


01sincos2 2 01sin)sin1(2 2 01sinsin22 2

01sinsin2 2 0)1sin2)(1(sin

2

1sinor 1sin

90 30360or 30180

330or 210 ,90


Solve the equation 2cos2 sin 1 0 for 0 360.

Solution:

Transform the equation into a quadratic equation with sin as the unknown.

01sinsin2 2

P. 32

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.

Consider y sin x. For every angle x, there is a corresponding trigonometric ratio y. Thus, y is a function of x.

1144 ..66 Graphs of Trigonometric Graphs of Trigonometric FunctionsFunctions

AA. . The Graph of y The Graph of y sin x sin x

x 0 30 60 90 120 150 180

y 0 0.5 0.87 1 0.87 0.5 0

From the above table, we can plot the points on the coordinate plane.

x 210 240 270 300 330 360

y 0.5 0.87 1 0.87 0.5 0

P. 33

The graph of y sin x repeats itself in the intervals –360 x 0, 0 x 360, 360 x 720, etc.

We can also plot the graph of y sin x for 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.

AA. . The Graph of y The Graph of y sin x sin x

P. 34


BB. . The Graph of y The Graph of y cos x 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.

P. 35


BB. . The Graph of y The Graph of y cos x 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. 36


CC. . The Graph of y The Graph of y tan x tan x

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.

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.

P. 37



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

P. 38



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:

2. y tan x is a periodic function with a period of 180.

From 0 to 90, tan x increases from 0 to positive infinity.From 90 to 180, tan x increases from negative infinity to 0.

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. 39

For example, to find the maximum and minimum values of 3 4cos x:

Given a trigonometric function, we can find its maximum and minimum values algebraically.


1 cos x 1 4 4cos x 4

4 3 3 4cos x 4 31 3 4cos x 7

The maximum and minimum values are 7 and 1 respectively.


P. 40

Now, we will study the transformations on the graphs of trigonometric functions.

In Book 4, we learnt the transformations such as translation and reflection of graphs of functions.


DD. . Transformation on the Graphs of Transformation on the Graphs of Trigonometric FunctionsTrigonometric Functions

P. 41

Example 14.10T

DD. . Transformation on the Graphs of Transformation on the Graphs of Trigonometric FunctionsTrigonometric Functions

(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

Solution:


(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.

y cos x 2

y cos (x 180)y cos x

P. 42

We should note that the graphical solutions are approximate in nature.

Similar to quadratic equations, trigonometric equations can be solved either by the algebraic method or the graphical method.

1144 ..77 Graphical Solutions of Graphical Solutions of Trigonometric EquationsTrigonometric Equations

P. 43

Example 14.11T

Solution:


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

(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.

y 0.6

y 0.7

P. 44

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:

(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.

y 1.5

P. 45

In a rectangular coordinate plane, the x-axis and the y-axis divide the plane into four quadrants.

Chapter Chapter SummarySummary14.1 Introduction to Trigonometry

P. 46

Chapter Chapter SummarySummary

x

ytan

r

xcos

r

ysin

The signs of different trigonometric ratios in different quadrants can be memorized by the ASTC diagram.

14.2 Trigonometric Ratios of Arbitrary Angles

P. 47

If is the reference angle of an angle , then sin sin ,

cos cos , tan tan ,

where the choice of the sign ( or ) depends on the quadrant in which lies.

Chapter Chapter SummarySummary14.3 Finding Trigonometric Ratios Without

Using a Calculator

P. 48


1. (a) sin (180 – ) sin (b) cos (180 – ) –cos (c) tan (180 – ) –tan

2. (a) sin (180 ) –sin (b) cos (180 ) –cos (c) tan (180 ) tan

3. (a) sin (360 – ) –sin (b) cos (360 – ) cos (c) tan (360 – ) –tan

14.4 Trigonometric Identities

P. 49

Trigonometric equations can be solved by the algebraic method.

Chapter Chapter SummarySummary14.5 Trigonometric Equations

P. 50

1. Graph of y sin x


2. Graph of y cos x

3. Graph of y tan x

5. The periods of sin x, cos x and tan x are 360, 360 and 180 respectively.

4. For any real value of x, 1 sin x 1 and 1 cos x 1.

14.6 Graphs of Trigonometric Functions

P. 51

Trigonometric equations can be solved by the graphical method.

Chapter Chapter SummarySummary14.7 Graphical Solutions of Trigonometric Equations

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Trigonometry (1)