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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