Term 3 : Unit 1Trigonometry (Part B)

Name : ____________ ( ) Class : ______ Date :________

1.3 Simple Identities

1.4 Trigonometric Equations

Simple Trigonometric Identities and Equations

1.3 Simple Identities

In this lesson, we will

• define the secant, cosecant and cotangent functions,

• learn some simple trigonometric identities.

Objectives

Trigonometric Ratios of Acute AnglesThe three

trigonometric ratios are defined

as

OPQ is a right angled triangle.

adjacent

oppositehypotenuse

cosec

sec

cot

oppositeoppositehypotenusehypotenuse

adjacentadjacent

1

sin

1

cos

1

tan

Simple Trigonometric Identities and Equations

O

y

x

P (x, y)

r

Consider angles in the Cartesian plane.

Simple Trigonometric Identities and Equations

tany

x

yrxr

sintan

cos

2 22 2sin cos

y x

r r

2 2

2

y x

r

2

2

r

r

2 2sin cos 1 For any value

of θ.

r2 = x2 + y2

Simple Trigonometric Identities and Equations

1cot

tan

sin

cos

1

coscot

sin

2 2sin cos 1 2 2sin cos 1 2 2

2 2 2

sin cos 1

cos cos cos

2 2

2 2 2

sin cos 1

sin sin sin

2 2tan 1 sec 2 21 cot cosec

Simple Trigonometric Identities and Equations

Given that 2sin and cos 1, show thatx y 22 4 1 4x y

2 2sin cos 1

22 4 1 4x y

2sin sin2

xx

cos 1 cos 1y y

2

21 1

2

xy

From the identity

Rearranging

Rearranging

Example 3

Simple Trigonometric Identities and Equations

21 cosec

Simplify .1 sin 1 sin

x

x x

2cosec x

21 cosec

1 sin 1 sin

x

x x

2

2

cot

1 sin

x

x

2

1

sin x

2

2

cossin

cos

xxx

Rearranging 1 + cot2x = cosec2x

Using the identities

Cancelling

Example 1

Simple Trigonometric Identities and Equations

Prove the following identity sec cos sin tan .x x x x

sin tanx x

sec cosx x 1cos

cosx

x

2sin

cos

x

x

21 cos

cos

x

x

Using the identity

sinsin

cos

xx

x

Example 2

Simple Trigonometric Identities and Equations

Given that cosec cot 3, evaluate cosec cot and cos .A A A A A

cosec cot cosec cot 3 cosec cotA A A A A A

Using the identity

1 + cot2x = cosec2x.

cosec cot 3A A

2 2cosec cot 3 cosec cotA A A A 1 3 cosec cotA A

13cosec cotA A

13cosec cot cosec cot 3A A A A

1032cosecA 53cosecA

1sin

cosecA

A

2351

35

452cos 1 sinA A

Example 3

Simple Trigonometric Identities and Equations

2 2Show that sin can be expressed as cot and

1 cos 1 cos

find the value of .

k

k

2 2sin

1 cos 1 cos

4cos

sin

2

2 2cos 2 2cossin

sin

Using the identity.4cot

2

2 1 cos 2 1 cossin

1 cos

Using the identity.

Example 4

4k

Simple Trigonometric Identities and Equations

1.4 Trigonometric Equations

In this lesson, we will solve some further trigonometric equations by

simplifying or factorising, to reduce them to the form sin x = k, cos x = k and

tan x = k.

Objectives

O

y

x

AS

T C

x

Find all the angles between 0° and 360° which satisfy the equation 3 cos x + 2 sin x = 0.

Simple Trigonometric Identities and Equations

3cos 2sin 0x x

2sin 3cosx xsin 3

cos 2

x

x

56.3

123.7 , 303.7x , 0180 36x

cos x ≠ 0

3tan

2x

3tan

2

tan x < 0 so x is in the 2nd or the 4th quadrant.

Using the identity.

Calculate the base angle α.

Example 5

O

y

x

AS

T C

y

Find all the angles between 0o and 360o which satisfy the equation sin y = 4 tan y.

Simple Trigonometric Identities and Equations

sin 4 tan , 0 360o oy y y sin

sin 4cos

yy

y

180oy sin 0y

sin cos 4siny y ysin cos 4sin 0y y y

Using the identity

Factorise, do not cancel

through by sin θ. sin cos 4 0y y

cos 4y No solutions –1 ≤ θ ≤ 1

Example 6

O

y

x

AS

T C

y

= 30°

Find all the angles between 0° and 360° which satisfy the equation 2 cos2 y – 1 = sin y.

Simple Trigonometric Identities and Equations

22cos 1 sin , 0 360y y y 22 1 sin 1 siny y

270y , 130 50y

Using sin2y + cos2y =

1

22 2sin 1 siny y

sin y > 0 so y is in the 1st or the 2nd quadrant.

Factorising

22sin sin 1 0y y 2sin 1 sin 1 0y y

12sin y sin 1y

O

y

x

AS

T C

y

Example 7

O

y

x

AS

T C

x + 30°

Find all the angles between 0° and 360° which satisfy the equation cos (x + 30o) = – 0.3.

Simple Trigonometric Identities and Equations

cos 30 0.3x

0 360x 30 30 390x

72.5

30 107.5 , 252.5x

30 18 , 1800x cos 0.3

cos (x + 30°) < 0 so x is in the 2nd or the 3rd quadrant. Calculate the

basic angle α.

72.5 , 18030 180 72.5x

77.5 , 222.5x

Example 8

O

y

x

AS

T C

2x

Find all the angles between 0° and 360° which satisfy the equation sin 2x = 0.866.

Simple Trigonometric Identities and Equations

sin 2 0.866x

0 360x 0 2 720x

60

20 , 4260 , 1 0 , 480

2 , 360, , 540180x sin 0.866

sin 2x > 0 so x is in the 1st or the 2nd quadrant. Calculate the

basic angle α.

, 366 0 , 5400 , 180 60 60 60

30 , 60 , 210 , 240x

Example 9