What is the general solution for sin^2(x)=((6m-5))/6 ?

The general solution for sin^2(x)=((6m-5))/6 is

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sin^2(x)=((6m-5))/6

sin^{2} (x) = \frac{( 6 m - 5 )}{6}

x = arcsin (\frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (\frac{6 m - 5}{6}) + 2 πn

Solution steps

sin^{2} (x) = \frac{( 6 m - 5 )}{6}

Solve by substitution

sin^{2} (x) = \frac{6 m - 5}{6}

Let:

sin (x) = u

u^{2} = \frac{6 m - 5}{6}

u^{2} = \frac{6 m - 5}{6} : u = \frac{6 m - 5}{6}, u = - \frac{6 m - 5}{6}

u^{2} = \frac{6 m - 5}{6}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{6 m - 5}{6}, u = - \frac{6 m - 5}{6}

Substitute back

u = sin (x)

sin (x) = \frac{6 m - 5}{6}, sin (x) = - \frac{6 m - 5}{6}

sin (x) = \frac{6 m - 5}{6}, sin (x) = - \frac{6 m - 5}{6}

sin (x) = \frac{6 m - 5}{6} : x = arcsin (\frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (- \frac{6 m - 5}{6}) + 2 πn

sin (x) = \frac{6 m - 5}{6}

Apply trig inverse properties

sin (x) = \frac{6 m - 5}{6}

General solutions for

sin (x) = \frac{6 m - 5}{6}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (\frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (- \frac{6 m - 5}{6}) + 2 πn

x = arcsin (\frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (- \frac{6 m - 5}{6}) + 2 πn

sin (x) = - \frac{6 m - 5}{6} : x = arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (\frac{6 m - 5}{6}) + 2 πn

sin (x) = - \frac{6 m - 5}{6}

Apply trig inverse properties

sin (x) = - \frac{6 m - 5}{6}

General solutions for

sin (x) = - \frac{6 m - 5}{6}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (\frac{6 m - 5}{6}) + 2 πn

x = arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (\frac{6 m - 5}{6}) + 2 πn

Combine all the solutions

x = arcsin (\frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = arcsin (- \frac{6 m - 5}{6}) + 2 πn, x = π + arcsin (\frac{6 m - 5}{6}) + 2 πn

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

sin (x) + cos (\frac{π}{2} - x) = 1

4 sin^{2} (x) + 2 sin (x) = 2

5 cos (x) tan (x) - 3 tan (x) = 0

csc (θ) = - \frac{3}{5}

What is the general solution for sin^2(x)=((6m-5))/6 ?
The general solution for sin^2(x)=((6m-5))/6 is