What is the general solution for sin(2θ)=-24/25 ,sin(θ)= 3/5 ?

The general solution for sin(2θ)=-24/25 ,sin(θ)= 3/5 is No Solution for θ\in\mathbb{R}

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sin(2θ)=-24/25 ,sin(θ)= 3/5

sin (2 θ) = - \frac{24}{25}, sin (θ) = \frac{3}{5}

No Solution for θ \in R

Solution steps

sin (2 θ) = - \frac{24}{25}, sin (θ) = \frac{3}{5}

Apply trig inverse properties

sin (2 θ) = - \frac{24}{25}

General solutions for

sin (2 θ) = - \frac{24}{25}

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

2 θ = arcsin (- \frac{24}{25}) + 2 πn, 2 θ = π + arcsin (\frac{24}{25}) + 2 πn

2 θ = arcsin (- \frac{24}{25}) + 2 πn, 2 θ = π + arcsin (\frac{24}{25}) + 2 πn

Solve

2 θ = arcsin (- \frac{24}{25}) + 2 πn : θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn

2 θ = arcsin (- \frac{24}{25}) + 2 πn

Simplify

arcsin (- \frac{24}{25}) + 2 πn : - arcsin (\frac{24}{25}) + 2 πn

arcsin (- \frac{24}{25}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{24}{25}) = - arcsin (\frac{24}{25})

= - arcsin (\frac{24}{25}) + 2 πn

2 θ = - arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

2 θ = - arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = - \frac{arcsin ( \frac{24}{25} )}{2} + \frac{2 πn}{2}

Simplify

θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn

θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn

Solve

2 θ = π + arcsin (\frac{24}{25}) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

2 θ = π + arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

2 θ = π + arcsin (\frac{24}{25}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

θ = - \frac{arcsin ( \frac{24}{25} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{24}{25} )}{2} + πn

Solutions for the range

sin (θ) = \frac{3}{5}

No Solution for θ \in R

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What is the general solution for sin(2θ)=-24/25 ,sin(θ)= 3/5 ?
The general solution for sin(2θ)=-24/25 ,sin(θ)= 3/5 is No Solution for θ\in\mathbb{R}