Chapter 5 DM015 TRIGONOMETRY 2021.2022

DM015 CHAPTER 5 2021/2022

CHAPTER 5

TRIGONOMETRY

5.1 ANGLES AND THEIR MEASURES

Radian and Degree Measures of a Angle

Definition
The angle 1 radian is defined as an angle subtended by an arc of length at the
centre 0 of a circle with radius .

 radians = 1800 rr

1 radian

ran

 180 0    

1 radian =  or 1° = 180 0 radians

Converting between Degree and Radian Measure

➢ To convert from degree to radian measure, multiply by  radians .
180

➢ To convert from radian to degree measure, multiply by 180
 radians .

➢ 1 = 60' (60 minutes)

➢ 1' = 60'' (60 seconds)

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EXAMPLE 1:

1. Express each of the following angles in degrees.

(a) 2.45 radians (b) 3.54 radians

(c) 2 radians (d) 7 radians

3 4

(e) 0.694 radians (f) 5 radians

2. Express the following in radians as exact values in terms of .

(a) 210° (b) 450°

(c) −105° (d) −405°

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3. Convert each of the following angles to radians.

(a) 75° (b) 136°15′

(c) 82.36° (d) 16°30′15′′

The equivalents radian-degree

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Trigonometric Ratio for Any Angle

(i) All angle is positive when it is measured anti clockwise from the
positive -axis and is negative when it is measured clockwise

Positive angle

Negative angle

(ii) Signs of trigonometric ratios
A reference angle for an angle is the positive acute angle made by
the terminal side of angle and the -axis. (Shown below in red)

Quadrant II Quadrant I

sin + All +

tan + cos +

Quadrant III Quadrant IV

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(iii) Negative Angle

Irrespectively of whether is an acute angle, obtuse angle or reflex

angle, if is a negative angle,

(− ) = − ( )
(− ) =
(− ) = −

(iv) Types of Angles

 = 0 0    90 90   180

 = 90  =180

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180    360  = 360

4. State the trigonometric ratio in acute angle.

(a) sin 160° (b) cos 220°

(c) tan 310° (d) cos 172°

(e) tan 246° (f) sin 325°

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5. State the trigonometric ratio in acute angle

(a) sin(−25°) (b) cos(−40°)

(c) tan(−48°) (d) sin(−128°)

(e) cos(−152°) (f) tan(−163°)

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5.2 TRIGONOMETRIC FUNCTIONS

Trigonometric Ratios of , , , , and .

sin = opposite SOH

hypotenuse A

cos = adjacent hypotenuse opposite

hypotenuse CAH

tan = opposite B adjacent C

adjacent TOA

From the diagram above, we can define trigonometric ratios for the sides as:

Reciprocal Identities


= =


= =


= =


= =

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A

z 90 −

y

C

Bx

sin(90° − ) = = cos cosec(90° − ) = sec
sec(90° − ) = cosec
cot(90° − ) = tan
cos(90° − ) = = sin


tan(90° − ) = = cot


EXAMPLE 2:

1. If cot = 3 and is an acute angle, find

4,

(a) tan (b) sin

(c)sec (d)cosec

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2. Given cos = 0.8, where 0° ≤ ≤ 90° evaluate

5 sin − 3 tan + 3 cosec .

3. Given tan = 1 , where 0° ≤ ≤ 90° find

√7

cosec2 −sec2

(a) cosec2 +sec2

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(b)
sin2
4+cos2

4. Given that tan = √5 and is an acute angle, find without the use of table
2

or a calculator,

(a) cot (b) sin(90° − )

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(c) sec(90° − ) (d) cosec(90° − )

5. Given that cosec = 3√2 and is an acute angle, find without the use of
4

table or a calculator,

(a) tan (b) cot (90° − )

(c) cos (90° − ) (d) sec(90° − )

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Trigonometric Ratios of Particular Angles

Equilateral triangle of sides unit in length

22

1 1

Isosceles triangle

45o
1

45o
1

Equilateral triangle and Isosceles triangle can be used to find the
trigonometric ratios of particular angels.

 0 30 45 60 90
0 1
1 11 3
0 0
2 22

31 1

2 22

1 3

1

3

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(b) cos 210°
6. Find the exact value of
(a) sin 120°

(c) tan 225° (d) tan(−120°)

(e) sin(−135°) (f) cos(−270°)

7. Find the exact value of (b) cos 630°
(a) sin 390°

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8. Find the exact value of,

(a) tan  2  (b) sin  −  
 3   6 
 

(c) cos  11  (d) cos  11 
 3   3 
   

9. Prove that cos 60° = 1−tan2 30°
1+tan2 30°

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10. Prove that sec 30° tan 60° + sin 45° cosec 45° + cos 30° cot 60° = 7

.
2

11. Find the exact value of each expression. Do not use calculator.

(a) sin 90° + tan 45° (b) cos 180° − sin 180°

(c) cosec 45° tan 60° (d) sec 30° cot 45°

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(e) sin − cos (f) 3 cosec + cot

44 34

12. If sin = 12 where < 90°, find cos and tan without using the
13

calculator.

13. If tan = − 15 where 270° < < 360°, find cos and sin without using
8

the calculator.

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14. If sin = −185 in a third quadrant, find sec and tan .

15. If sec  = 5 in a fourth quadrant, find sin  and tan  .
3

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5.3 TRIGONOMETRIC IDENTITIES

Use basic trigonometric identities

cos2 + sin2 = 1
1 + tan2 = sec2
1 + cot2 = cosec2

Useful formulae and identities are summarized in the following diagram. The
diagonals show the reciprocals of the various trigonometric ratios:

sin = 1 cos = 1 tan = 1
cos ec sec cot

EXAMPLE 3: tan cot − cos2 = sin2
1. Prove that

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2. Prove that
cos +sin tan = sec

3. Prove that (sec + tan )(sec − tan ) =1

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4. Prove that
(cosec + cot )(cosec − cot ) =1

5. Prove the identity tan + cot = sec cosec .

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6. Prove the identity
tan xsec x = sin x .
1+ tan2 x

7. Prove the identity 1 + 1 − 1 = 2cos ec2 .
1+ cos cos

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8. Prove the identity
( )cosec2 x tan2 x −sin2 x = tan2 x

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9. Prove the identity
sec = cosec2
sec − cos

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5.4 TRIGONOMETRIC FORMULAE

Use Compound angle formulae

sin( ± ) = sin cos ± cos sin
cos( ± ) = cos cos ∓ sin sin
tan( ± ) = tan ±tan

1∓tan tan

EXAMPLE 4:

1. Without using calculator, find the value for the following in terms of surd.

(a) sin(45° + 30°) (b) cos(150° − 45°)

(c) tan(45° − 30°) (d) tan (1050 )

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2. Express the following in a single term.
(a) sin 2 cos + cos 2 sin

(b) cos 2 cos 3 + sin 2 sin 3

tan −1

(c) 1+tan

3. Prove the following identities

(a) sin( + ) = tan + tan

cos cos

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(b) (cos + cos )2 + (sin + sin )2 = 2[1 + cos( − )]

(c) tan ( x + 60) tan ( x − 60) = tan2 x −3
1−3tan2 x

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4. Find the values of sin( + ), cos( + ), tan( + ), if given:

(a) sin = 3 cos = 5 and in first quadrant.

5, 12 13, 7

(b) cos = − 13, tan = 24, is second quadrant and in third

quadrant.

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Use double-angle formulae

sin 2 = 2 sin cos

cos 2 = cos2 − sin2
= 2 cos2 − 1
= 1 − 2sin2

tan 2 = 2 tan
1−tan2

2 tan 15°

5. Find the exact value of 1−tan2 15°

6. If tan = 3 with is acute angle, find the exact value of:
4

(a) tan 2 (b) tan 4

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7. Without using calculator find the exact value of;

(a) 2 sin(22.5) cos(22.5) (b) 1 − 2 sin2(75)

8. Prove each of the following: (b) sin 3 = 3 sin − 4 sin3
(a) cot + tan 2 = cot sec 2

9. Given that is acute and that tan = 12, evaluate each of the following;

(a) tan 2 (b) sin 2

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(c) sec 2 (d) cosec 2

10. Prove that 1+ sin 2A = (sin A + cos A)2 .

11. Prove that sin 2 = tan , hence find the value of tan15 and tan 67.5
1+ cos 2

without using calculator.

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Solving of Trigonometric Equations

Solve basic trigonometric equations

Solve Equations such as sin = , cos = , tan = .

12. Find the value of , if 0° ≤ ≤ 360°

(a) sin = 1 (b) cos = 1

2 2

(c) sin = − √3 (d) tan = −2
2

(e) cos = − 1 (f) sin = − 1

4 4

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13. Solve the following equations for angles in the range 0° ≤ ≤ 360°

(a) tan 2 = √3 (b) sin 2 = − 1

2

(c) cos(3 − 75°) = 0.5 (d) cos 1 = −0

3

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14. Solve the following equations for values of for 0° ≤ ≤ 360°:

(a) 6 cos2 + cos − 2 = 0

(b) 2 sin2 − sin − 1 = 0

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(c) sec2 x + tan x =1

(d) 10sec2 x −11tan x =16

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(e)
sec x = 13 − tan2 x +16
sec x

(f) 5 + tan2 x = 9 − sec x
sec x

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15. Use Trigonometric identities to solve these questions in range

−180  x 180

(a) cosec2x = 3cot x −1

(b) 2sin2 x + 5cos x +1 = 0

(c) tan2 x + sec2 x =17

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(d)
2cos x + tan x = sec x

(e) cos xcot2 x = cos x

(f) sin x + 2cos2 x =1

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16. Solve the equations tan = 2 sin in radian, where 0° ≤ ≤ 2 .

17. Solve the following trigonometric equations

(a) 4cot2 x −9cosec x + 6 = 0 0° ≤ ≤ 360°

(b) 2tan2 x =11sec x −7 0° ≤ ≤ 360°

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(c) 3cot2 + 5 cosec + 1 = 0 0° ≤ ≤ 2

(d) 4 cos2 = 3 sin2 0° ≤ ≤ 360°

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