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

The general solution for 2sin^2(x)=3cos^2(x) is x=-0.88607…+pin,x=0.88607…+pin

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2sin^2(x)=3cos^2(x)

2 sin^{2} (x) = 3 cos^{2} (x)

x = - 0.88607 \dots + πn, x = 0.88607 \dots + πn

Solution steps

2 sin^{2} (x) = 3 cos^{2} (x)

Subtract

3 cos^{2} (x)

from both sides

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

Factor

2 sin^{2} (x) - 3 cos^{2} (x) : (2 sin (x) + 3 cos (x)) (2 sin (x) - 3 cos (x))

(2 sin (x) + 3 cos (x)) (2 sin (x) - 3 cos (x)) = 0

Solving each part separately

2 sin (x) + 3 cos (x) = 0 or 2 sin (x) - 3 cos (x) = 0

2 sin (x) + 3 cos (x) = 0 : x = arctan (- \frac{3}{2}) + πn

2 sin (x) - 3 cos (x) = 0 : x = arctan (\frac{3}{2}) + πn

Combine all the solutions

x = arctan (- \frac{3}{2}) + πn, x = arctan (\frac{3}{2}) + πn

Show solutions in decimal form

x = - 0.88607 \dots + πn, x = 0.88607 \dots + πn

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

4 cos (2 θ) = 2 sin (θ) + 3

tan^{2} (x) - \frac{1}{3} = 0

sin (x) + cos^{2} (x) = 1

sin^{2} (x) - 1 = 0, 0 \leq x < 2 π

What is the general solution for 2sin^2(x)=3cos^2(x) ?
The general solution for 2sin^2(x)=3cos^2(x) is x=-0.88607…+pin,x=0.88607…+pin