What is the general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 ?

The general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 is x= pi/4+pin

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(6cos(x)+5sin(x))^2+11sin^2(x)=66

(6 cos (x) + 5 sin (x))^{2} + 11 sin^{2} (x) = 66

x = \frac{π}{4} + πn

Solution steps

(6 cos (x) + 5 sin (x))^{2} + 11 sin^{2} (x) = 66

Subtract

66

from both sides

36 cos^{2} (x) + 60 cos (x) sin (x) + 36 sin^{2} (x) - 66 = 0

Rewrite using trig identities

- 30 + 60 \cdot \frac{sin ( 2 x )}{2} = 0

60 \cdot \frac{sin ( 2 x )}{2} = 30 sin (2 x)

- 30 + 30 sin (2 x) = 0

Move

30

to the right side

30 sin (2 x) = 30

Divide both sides by

30

sin (2 x) = 1

General solutions for

sin (2 x) = 1

2 x = \frac{π}{2} + 2 πn

Solve

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

- 3 sin (B) - 2 = sin (B) - 2

sin (\frac{θ}{2}) = 2 - sin (\frac{θ}{2})

2 sin (x) cos (x) = cos (2 x)

(tan (x) + 1) (2 sin (x) - 1) = 0

sin (θ) = \frac{0.3924}{cos ( θ )}

What is the general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 ?
The general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 is x= pi/4+pin