sin^{22}(x)= 1/2

Solution

sin^{22} (x) = \frac{1}{2}

x = 1.32108 \dots + 2 πn, x = π - 1.32108 \dots + 2 πn, x = - 1.32108 \dots + 2 πn, x = π + 1.32108 \dots + 2 πn

Degrees

x = 75.69275 \dots^{\circ} + 36 0^{\circ} n, x = 104.30724 \dots^{\circ} + 36 0^{\circ} n, x = - 75.69275 \dots^{\circ} + 36 0^{\circ} n, x = 255.69275 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin^{22} (x) = \frac{1}{2}

Solve by substitution

sin^{22} (x) = \frac{1}{2}

Let:

sin (x) = u

u^{22} = \frac{1}{2}

u^{22} = \frac{1}{2}

For

x^{n} = f (a)

, n is even, the solutions are

Substitute back

u = sin (x)

Apply trig inverse properties

General solutions for

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

Apply trig inverse properties

General solutions for

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

Combine all the solutions

Show solutions in decimal form

x = 1.32108 \dots + 2 πn, x = π - 1.32108 \dots + 2 πn, x = - 1.32108 \dots + 2 πn, x = π + 1.32108 \dots + 2 πn