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sin^2(x)= 1/(sqrt(2))

Solution

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

x = 0.99893 \dots + 2 πn, x = π - 0.99893 \dots + 2 πn, x = - 0.99893 \dots + 2 πn, x = π + 0.99893 \dots + 2 πn

Degrees

x = 57.23490 \dots^{\circ} + 36 0^{\circ} n, x = 122.76509 \dots^{\circ} + 36 0^{\circ} n, x = - 57.23490 \dots^{\circ} + 36 0^{\circ} n, x = 237.23490 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

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

u^{2} = \frac{1}{2} : u = \frac{2}{2}, u = - \frac{2}{2}

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

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{2}{2}, u = - \frac{2}{2}

Substitute back

u = sin (x)

sin (x) = \frac{2}{2}, sin (x) = - \frac{2}{2}

sin (x) = \frac{2}{2}, sin (x) = - \frac{2}{2}

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 0.99893 \dots + 2 πn, x = π - 0.99893 \dots + 2 πn, x = - 0.99893 \dots + 2 πn, x = π + 0.99893 \dots + 2 πn

3 tan^{2} (θ) + 5 tan (θ) = 2

cos^{2} (x) + 4 cos (x) = - 3

tan (θ) = \frac{11}{9}

tan (θ) = \frac{11}{5}

tan^{2} (x) \cdot sin (x) = sin (x)