What is the general solution for sin(2θ)=(-3)/4 ,(3pi)/2 <= 2θ<= 2pi ?

The general solution for sin(2θ)=(-3)/4 ,(3pi)/2 <= 2θ<= 2pi is No Solution for θ\in\mathbb{R}

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sin(2θ)=(-3)/4 ,(3pi)/2 <= 2θ<= 2pi

sin (2 θ) = \frac{- 3}{4}, \frac{3 π}{2} \leq 2 θ \leq 2 π

No Solution for θ \in R

Solution steps

sin (2 θ) = \frac{- 3}{4}, \frac{3 π}{2} \leq 2 θ \leq 2 π

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

sin (2 θ) = - \frac{3}{4}

Apply trig inverse properties

sin (2 θ) = - \frac{3}{4}

General solutions for

sin (2 θ) = - \frac{3}{4}

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

2 θ = arcsin (- \frac{3}{4}) + 2 πn, 2 θ = π + arcsin (\frac{3}{4}) + 2 πn

2 θ = arcsin (- \frac{3}{4}) + 2 πn, 2 θ = π + arcsin (\frac{3}{4}) + 2 πn

Solve

2 θ = arcsin (- \frac{3}{4}) + 2 πn : θ = - \frac{arcsin ( \frac{3}{4} )}{2} + πn

2 θ = arcsin (- \frac{3}{4}) + 2 πn

Simplify

arcsin (- \frac{3}{4}) + 2 πn : - arcsin (\frac{3}{4}) + 2 πn

arcsin (- \frac{3}{4}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{4}) = - arcsin (\frac{3}{4})

= - arcsin (\frac{3}{4}) + 2 πn

2 θ = - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

2 θ = - arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = - \frac{arcsin ( \frac{3}{4} )}{2} + \frac{2 πn}{2}

Simplify

θ = - \frac{arcsin ( \frac{3}{4} )}{2} + πn

θ = - \frac{arcsin ( \frac{3}{4} )}{2} + πn

Solve

2 θ = π + arcsin (\frac{3}{4}) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( \frac{3}{4} )}{2} + πn

2 θ = π + arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

2 θ = π + arcsin (\frac{3}{4}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( \frac{3}{4} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} + \frac{arcsin ( \frac{3}{4} )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( \frac{3}{4} )}{2} + πn

θ = - \frac{arcsin ( \frac{3}{4} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{3}{4} )}{2} + πn

Solutions for the range

\frac{3 π}{2} \leq 2 θ \leq 2 π

No Solution for θ \in R

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What is the general solution for sin(2θ)=(-3)/4 ,(3pi)/2 <= 2θ<= 2pi ?
The general solution for sin(2θ)=(-3)/4 ,(3pi)/2 <= 2θ<= 2pi is No Solution for θ\in\mathbb{R}