The document provides information on solving more difficult trigonometric equations. It discusses finding the acute angle and related angles, using trig function properties to isolate the angle, and considering additional solutions that occur every period of the trig function. Example problems are worked through step-by-step to demonstrate these concepts.

1. Block 3
Solving Trig Equations

2. What is to be learned?
• How to solve more difficult trig equations

3. Find acute
angle then
get other
relations
Usually have two solutions
The solutions are related angles

4. NWD is vital
a0
180 – a
180 + a 360 - a
iii
iii iv
CT
AS
Sinx = 0.8
2nd
F/Inv/Shift
Sin 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

5. a0
180 – a
180 + a 360 - a
iii
iii iv
CT
AS
Sinx = -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

6. a0
180 – a
180 + a 360 - a
iii
iii iv
CT
AS
Tanx = 1.2
Tan-1
(1.2) = 500
+ve or –ve?
Tan +ve in i and iii
x = 500
or 180+50
= 2300

7. a0
180 – a
180 + a 360 - a
iii
iii iv
CT
AS
4cosx + 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 = -0.25
cos-1
(0.25) = 760
= 2560

8. sin 2x = 0.4

let 2x = A
becomes 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

9. sin 2x = 0.4

let 2x = A
becomes 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

10. cos 2x = 0.4

let 2x = A
becomes 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

11. Cos 2x = 0.4

let 2x = A
becomes 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

12. 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

13. a0180 – a
180 + a 360 - a
iii
iii iv CT
ASTrig Equations
so 2x + 30 = 127 or 233
2x + 30 = 127 or 2x + 30 = 233
2x = 97 2x = 203
x = 48.50
x = 101.50
to be continued

14. Additional Solutions
sinx = ½
x = 300
(or 1500
)
½
300
3900
7500
00
≤ x ≤ 3600
outwith limits
additional solutions occur every 3600
Period of sin x is 3600

15. sin 2x = 0.4

let 2x = A
becomes 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

16. sin 2x = 0.4
a0180 – 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

17. 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

18. back to ex
x = 48.50
x = 101.50
period = 1800
(360 ÷ 2)
additional solutions
+1800
228.50 281.50

19. sin (2x – 20) = -0.4
let 2x – 20 = A
becomes sin A = -0.4
sin-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