Solving trig equations higher

Block 3

Solving Trig Equations

What is to be learned?

• How to solve more difficult trig equations

Find acute angle then get other relations

Usually have two solutionsThe solutions are related angles

NWD is vital

a0180 – a

180 + a 360 - a

iii

iii iv

CT

ASSinx = 0.8

2ndF/Inv/ShiftSin 0.8

Angle?

Sin-1(0.8) = 530

Related acute angle

+ve or –ve?Sin +ve in i and ii

x = 530 or 180 – 53 = 1270

a0180 – a

180 + a 360 - a

iii

iii iv

CT

ASSinx = -0.4

Sin-1(0.4) = 240

+ve or –ve?Sin -ve in iii and iv

x = 180+24 or 360 – 24 = 2040

Always put a positive number here

= 3360

a0180 – a

180 + a 360 - a

iii

iii iv

CT

ASTanx = 1.2

Tan-1(1.2) = 500

+ve or –ve?Tan +ve in i and iii

x = 500 or 180+50 = 2300

a0180 – a

180 + a 360 - a

iii

iii iv

CT

AS4cosx + 2 = 1

4cosx = -1

+ve or –ve?cos -ve in ii and iii

x = 180 – 76 or 180+76 = 1040

Change to cosx=Get rid ofSubtract 2 from both sides

cosx = -0.25cos-1(0.25) = 760

= 2560

sin 2x = 0.4

let 2x = Abecomes sin A = 0.4

Sin-1(0.4) = 240

Sin +ve in i and ii A = 240 or 180 – 24

= 1560

a0180 – a

180 + a 360 - a

iii

iii iv CT

AS

sin 2x = 0.4

let 2x = Abecomes sin A = 0.4

Sin-1(0.4) = 240

Sin +ve in i and ii A = 240 or 180 – 24

= 1560

a0180 – a

180 + a 360 - a

iii

iii iv CT

AS

2x = 240 or 1560

x= 120 or 780

cos 2x = 0.4

let 2x = Abecomes cos A = 0.4

Cos-1(0.4) = 660

Cos +ve in i and iv A = 660 or 360 – 66

= 2940

a0180 – a

180 + a 360 - a

iii

iii iv CT

AS

Cos 2x = 0.4

let 2x = Abecomes Cos A = 0.4

Cos-1(0.4) = 660

Cos +ve in i and iv A = 660 or 360 – 66

= 2940

a0180 – a

180 + a 360 - a

iii

iii iv CT

AS

2x = 660 or 2940

x= 330 or 1470

cos (2x + 30) = -0.6

let 2x + 30 = A

becomes cos A = - 0.6

cos-1(0.6) = 530

Cos -ve in ii and iii A = 180 – 53

A = 1270 or 2330

a0180 – a

180 + a 360 - a

iii

iii iv CT

ASTrig Equations

or 180 + 53

so 2x + 30 = 127 or 233

a0180 – a

180 + a 360 - a

iii

iii iv CT

ASTrig Equations

so 2x + 30 = 127 or 233

2x + 30 = 127 or 2x + 30 = 2332x = 97 2x = 203x = 48.50 x = 101.50

to be continued

Additional Solutionssinx = ½

x = 300 (or 1500)½

300 3900 7500

00 ≤ x ≤ 3600

outwith limitsadditional solutions occur every 3600

Period of sin x is 3600

sin 2x = 0.4

let 2x = Abecomes sin A = 0.4

Sin-1(0.4) = 240

Sin +ve in i and ii A = 240 or 180 – 24

= 1560

a0180 – a

180 + a 360 - a

iii

iii iv CT

AS

2x = 240 or 1560

x= 120 or 780

sin 2x = 0.4a0180 – a

180 + a 360 - a

iii

iii iv CT

AS

x= 120 or 780

period = 1800

additional solutions every 1800

additional solutions 1920 or 2580

Additional Solutions

• The same solutions will reoccur in every trig graph cycle.

• Always check for additional solutions by adding (or subtracting) the periodsubtracting) the period

back to exx = 48.50 x = 101.50

period = 1800

(360 ÷ 2)

additional solutions+1800

228.50 281.50

sin (2x – 20) = -0.4let 2x – 20 = A

becomes sin A = -0.4sin-1(0.6) = 240

Sin -ve in iii and iv A = 180 + 24

A = 2040 or 3360

Key Question

or 360 – 24

so 2x – 20 = 2040 or 2x – 20 = 3360

x = 1120 or x = 1780

Additional solutions x = 2920 or 3580