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Popular >

sin^2(x)-2/3 =0

Solution

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

Solution

x = 0.95531 \dots + 2 πn, x = π - 0.95531 \dots + 2 πn, x = - 0.95531 \dots + 2 πn, x = π + 0.95531 \dots + 2 πn

+1

Degrees

x = 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 125.26438 \dots^{\circ} + 36 0^{\circ} n, x = - 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 234.73561 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

u^{2} - \frac{2}{3} = 0

u^{2} - \frac{2}{3} = 0 : u = \frac{2}{3}, u = - \frac{2}{3}

u^{2} - \frac{2}{3} = 0

Move

\frac{2}{3}

to the right side

u^{2} - \frac{2}{3} = 0

Add

\frac{2}{3}

to both sides

u^{2} - \frac{2}{3} + \frac{2}{3} = 0 + \frac{2}{3}

Simplify

u^{2} = \frac{2}{3}

u^{2} = \frac{2}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{2}{3}, u = - \frac{2}{3}

Substitute back

u = sin (x)

sin (x) = \frac{2}{3}, sin (x) = - \frac{2}{3}

sin (x) = \frac{2}{3}, sin (x) = - \frac{2}{3}

sin (x) = \frac{2}{3} : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

sin (x) = \frac{2}{3}

Apply trig inverse properties

sin (x) = \frac{2}{3}

General solutions for

sin (x) = \frac{2}{3}

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

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

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

sin (x) = - \frac{2}{3} : x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

sin (x) = - \frac{2}{3}

Apply trig inverse properties

sin (x) = - \frac{2}{3}

General solutions for

sin (x) = - \frac{2}{3}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 0.95531 \dots + 2 πn, x = π - 0.95531 \dots + 2 πn, x = - 0.95531 \dots + 2 πn, x = π + 0.95531 \dots + 2 πn

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