Solving Trig Functions

Question

Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)

cos 2𝜃 = 9 sin 𝜃 − 4

Answer 1

Theta = 30+360k, 150+360k

or pi/6 +2kpi, 5pi/6+2kpi

cos2T = 9sinT -4

1-2sin^2(T) = 9sinT -4

2sin^2(T) +9sinT -5

(2sinT-1)(sinT+5)=0

sinT=1/2 (ignore the other factor as sines are never less than -1)

T=sin^-1(1/2) = 30 or 150 degrees +360k degrees or pi/6 or5pi/6 + 2kpi

check the answers

cos60 = 9sin30 -4

1/2 = 9/2 -8/2

1/2 = 1/2

cos300=9sin150-4

1/2 =9(1/2) -4

1/2 = 1/2

Answer 2