What is the general solution for 6cos(2x)=3+5sin^2(2x) ?

The general solution for 6cos(2x)=3+5sin^2(2x) is x=(0.64350…)/2+pin,x=pi-(0.64350…)/2+pin

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6cos(2x)=3+5sin^2(2x)

6 cos (2 x) = 3 + 5 sin^{2} (2 x)

x = \frac{0.64350 \dots}{2} + πn, x = π - \frac{0.64350 \dots}{2} + πn

Solution steps

6 cos (2 x) = 3 + 5 sin^{2} (2 x)

Subtract

3 + 5 sin^{2} (2 x)

from both sides

6 cos (2 x) - 3 - 5 sin^{2} (2 x) = 0

Rewrite using trig identities

- 8 + 5 cos^{2} (2 x) + 6 cos (2 x) = 0

Solve by substitution

cos (2 x) = \frac{4}{5}, cos (2 x) = - 2

cos (2 x) = \frac{4}{5} : x = \frac{arccos ( \frac{4}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{4}{5} )}{2} + πn

cos (2 x) = - 2 :

No Solution

Combine all the solutions

x = \frac{arccos ( \frac{4}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{4}{5} )}{2} + πn

Show solutions in decimal form

x = \frac{0.64350 \dots}{2} + πn, x = π - \frac{0.64350 \dots}{2} + πn

2 + 2 sin (θ) = cos^{2} (θ)

2 cos (x) cos (2 x) - 2 sin (x) sin (2 x) = - 1

- 19 sin (θ) + \frac{19}{2} = 0

y, e^{8 x} = sin (x + 3 y)

π cos (π x) + π sin (π x) = 0

What is the general solution for 6cos(2x)=3+5sin^2(2x) ?
The general solution for 6cos(2x)=3+5sin^2(2x) is x=(0.64350…)/2+pin,x=pi-(0.64350…)/2+pin