What is the general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 ?

The general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 is x= pi/(12)+pin,x=(5pi)/(12)+pin

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(-6cos(x)-5sin(x))^2+11sin^2(x)=51

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Solution steps

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

Subtract

51

from both sides

36 cos^{2} (x) + 60 cos (x) sin (x) + 36 sin^{2} (x) - 51 = 0

Rewrite using trig identities

- 15 + 60 \cdot \frac{sin ( 2 x )}{2} = 0

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

- 15 + 30 sin (2 x) = 0

Move

15

to the right side

30 sin (2 x) = 15

Divide both sides by

30

sin (2 x) = \frac{1}{2}

General solutions for

sin (2 x) = \frac{1}{2}

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

Solve

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

Solve

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

cos (5 x) - sin (2 x) - cos (x) = 0

cosh (4 x) = 0

x, tan (2 x) = - \frac{( \frac{30 + 20}{2} )}{80}

4 cos (3 θ) = 2

sin (x) = - 0.8176

What is the general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 ?
The general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 is x= pi/(12)+pin,x=(5pi)/(12)+pin