Online Material Trigonomatryic 7.1 a _ d

Based on the following videos:

7.1 (a) State Trigonometric Ratios of Sine, Cosine, Tangent, Cosecant, Secant and Cotangent
7.1(b) Complementary angles
7.1 (c) Special Angles For Trigonometric Functions
7.1 (c) Evaluate Trigonometric Functions ( Positive & Negative Angle)
7.1 (d) Evaluate Trigonometric Functios For Any Angle

7.1 Trigonometric Ratios and Identities

LEARNING OUTCOMES :

At the end of the lesson, students should be able to :

a) state the trigonometric ratios of sin  , cos , tan  , cos ec ,

sec , and cot  . (C1 : Lecture)

b) use tan  sin
cos

sin90    cos

cos90    sin

 tan 90   cot (C1 : Lecture)

7.1 Trigonometric Ratios and Identities

LEARNING OUTCOMES :
At the end of the lesson, students should be able to :

c) use some special angles (C3 : Tutorial )

d) evaluate trigonometric functions for any angles (C2 : Lecture)

e) use the Pythagoras identities :  sin  cos 12 2
(C3 : Tutorial )

1 tan 2   sec2 

1 cot2   cos ec2

Introduction to Trigonometry

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Trigonometry ...
is all about triangles.

State Trigonometric ratios of sin , cos , tan , cosec , sec , and cot .

Let  be an acute angle of a right triangle. Then the 6 trigonometric
functions of  are as follows :

 P sin   opposite  PQ

R hypotenuse PR

cos  adjacent  QR

hypotenuse PR

Q tan  opposite  PQ

adjacent QR

Beecher, Penna, Bittinger, “ Algebra & Trigonometry”, Addison Wesley. 2002, p 337. 5

7.1 (b) Complimentary Angle

Two angles are complementary if their z y
sum is 90 . θ

a) sin(90  )  x x

z b) cos(90  )  y

cos  x z

z sin   y

z

Therefore sin(90  )  cos Therefore

cos(90  )  sin 

Barnett, Ziegler & Byleen, “Colleg Algebra with Trigonometry”, Mc Graw Hill, . 2001, p 416.. 6

7.1 (b) Complimentary Angle

c) tan(90  )  x z y

y θ
x
cot  x

y

therefore tan(90  )  cot

Example 1 (a)

If cot  3 and is an acute angle, find tan ,sin  ,sec and cosec.

4

Solution:

5 4 cot  3

 4

3 cot  1

tan

tan  4 sin   4 sec  5 cosec  5

3 5 3 4

8

Example 1 (b)

Given cos x  0.8 , evaluate 5sin x  3 tan x  3cos ec x.

Solution: 5sin x  3 tan x  3cosecx.
 5 3   3 3   3 5 
5
3 5 4 3

x  3 9  5  23  5 3
4 44
4

9

Example 1 (c)

Given tan  1 , find sin 2 
4  cos2 
7

a) b)

cos ec2  sec2 
cos ec2  sec2 

10

Solution :  cosec2 2 2 2   2 2 2
 a) cosec22 7
tan  1  sec2  
 sec2  2
7 2   2 2
2 7
22
8 8
1 7
 8

8  7
7
 56  8  48  3
56  8 64 4

11

22 sin 2    1 2  1
1 b) 4  cos2  2 2
8
 4   2 7 2 4 7
2
7 8

 1 1
32  7 39

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7.1 (c) Reference Angle / Basic Angle (α)

 Reference angle or basic angle is the acute angle formed by the
terminal side of the angle and the x-axis.

Sine positive All positive

II θ I
α =180o - θ θ
α=θ
α

α = θ - 180o α α α =360o - θ

III IV
Tangent positive θ

Cosine positive

7.1(d) Positive and Negative Angles
y

SA sin    sin



x cos   cos



TC tan   tan

A counter clockwise rotation produces a positive angle, and
a clockwise rotation produces a negative angle.

Barnett, Ziegler & Byleen, “Colleg Algebra with Trigonometry”, Mc Graw Hill, . 2001, p 414..
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Example 2 (a)

If tan   1 and  is in the fourth quadrant.

2

Find the sec and cosec .

Solution

Hence

2

θ sec  1  3
cos 2

-1 cos ec  1   3
3
sin 

15

Solution  cosec2 2 2 2   2 2 2
 a) cosec22 7
22  sec2  
 sec2  2
1 2   2 2
2 7


7

8 8
7
 8

8  7

 56  8  48  3
56  8 64 4

16

22 sin 2    1 2  1
1 b) 4  cos2  2 2
8
 4   2 7 2 4 7
2
7 8

 1 1
32  7 39

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