SM015 - 7.1 (Notes Solutions)

7.1 Trigonometric Ratios & Identities , &.

Friday, July 10, 2020 3:14 PM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) State trigonometric ratio of , , ,
(b) Express

(i)

(ii)
(iii)
(iv)
(c) Use some special angles.

(d) Find the angle of the trigonometric equations.
(e) Proof of the trigonometric identities.

Topic 7.1 Page 1

Trigonometry

Tuesday, May 26, 2020 10:21 PM

Trigonometric ratios
Sine :

Cosine :
Tangent :

Secant :
Cosecant :

Cotangent :

Relationship among trigonometric ratios

Type of angle
Acute angle
Right angle
Obtuse angle
Reflex angle

Topic 7.1 Page 2

Example 1 & 2 and .

Friday, May 29, 2020 10:16 PM

Example 1
If and is an acute angle, find

Solution:

Example 2 and
If and is in the fourth quadrant, find

.

Solution:

Topic 7.1 Page 3

Angle units

Tuesday, May 26, 2020 11:02 PM

 An angle can be measure using
(i) degree
(ii) radian

 The relationship between degree & radian:

 E.g.

=

 Angle commonly used: Radian
Degree

Topic 7.1 Page 4

Trigonometric Ratios for Special Angle

Thursday, May 28, 2020 2:59 PM

undefined undefined
for
Remarks:
Refer to graph of

Topic 7.1 Page 5

The graphs of sine, cosine & tangent

Thursday, May 28, 2020 4:23 PM

Further reference : https://www.desmos.com/calculator/zfbhdeyjsu

Topic 7.1 Page 6

Example 3 & 4 .
Alternative Solution: (From LHS to RHS)
Friday, May 29, 2020 10:16 PM

Example 3

Prove that
Solution: (From RHS to LHS)

Example 4 .
Prove that

Solution:

Topic 7.1 Page 7

Basic Angle ,

Thursday, May 28, 2020 4:40 PM

 an acute angle formed between x-axis and the line r

Topic 7.1 Page 8

Positive & Negative Angle

Thursday, May 28, 2020 5:09 PM

Positive Angle Negative Angle
 angle formed by anti-clockwise  angle formed by clockwise

rotation from rotation from

positive side of x-axis pivoted positive side of x-axis

at the origin pivoted at the origin

Remarks:

Example 5

Without using calculator, evaluate

Solution (a): Solution (b):

Topic 7.1 Page 9

Trigonometric Identity

Thursday, May 28, 2020 6:14 PM

By Pythagoras Theorem :
Divide with r 2 :
Divide with x 2 :
Divide with y 2 :

Topic 7.1 Page 10

Example 6

Friday, May 29, 2020 10:16 PM

Prove the following identities.
Solution:

[Shown]

Prove the following identities.
Solution:

[Shown]

Topic 7.1 Page 11

Example 7

Friday, May 29, 2020 10:16 PM

Prove the identity
Solution:

[Proven]
Alternative: (From RHS to LHS)

[Proven]

Topic 7.1 Page 12