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Popular >

6sin^2(θ)=2

Solution

6 sin^{2} (θ) = 2

Solution

θ = 0.61547 \dots + 2 πn, θ = π - 0.61547 \dots + 2 πn, θ = - 0.61547 \dots + 2 πn, θ = π + 0.61547 \dots + 2 πn

+1

Degrees

θ = 35.26438 \dots^{\circ} + 36 0^{\circ} n, θ = 144.73561 \dots^{\circ} + 36 0^{\circ} n, θ = - 35.26438 \dots^{\circ} + 36 0^{\circ} n, θ = 215.26438 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 sin^{2} (θ) = 2

Solve by substitution

6 sin^{2} (θ) = 2

Let:

sin (θ) = u

6 u^{2} = 2

6 u^{2} = 2 : u = \frac{1}{3}, u = - \frac{1}{3}

6 u^{2} = 2

Divide both sides by

6

6 u^{2} = 2

Divide both sides by

6

\frac{6 u ^{2}}{6} = \frac{2}{6}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = sin (θ)

sin (θ) = \frac{1}{3}, sin (θ) = - \frac{1}{3}

sin (θ) = \frac{1}{3}, sin (θ) = - \frac{1}{3}

sin (θ) = \frac{1}{3} : θ = arcsin (\frac{1}{3}) + 2 πn, θ = π - arcsin (\frac{1}{3}) + 2 πn

sin (θ) = \frac{1}{3}

Apply trig inverse properties

sin (θ) = \frac{1}{3}

General solutions for

sin (θ) = \frac{1}{3}

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

θ = arcsin (\frac{1}{3}) + 2 πn, θ = π - arcsin (\frac{1}{3}) + 2 πn

θ = arcsin (\frac{1}{3}) + 2 πn, θ = π - arcsin (\frac{1}{3}) + 2 πn

sin (θ) = - \frac{1}{3} : θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

sin (θ) = - \frac{1}{3}

Apply trig inverse properties

sin (θ) = - \frac{1}{3}

General solutions for

sin (θ) = - \frac{1}{3}

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

θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{1}{3}) + 2 πn, θ = π - arcsin (\frac{1}{3}) + 2 πn, θ = arcsin (- \frac{1}{3}) + 2 πn, θ = π + arcsin (\frac{1}{3}) + 2 πn

Show solutions in decimal form

θ = 0.61547 \dots + 2 πn, θ = π - 0.61547 \dots + 2 πn, θ = - 0.61547 \dots + 2 πn, θ = π + 0.61547 \dots + 2 πn

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