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

The general solution for sin(θ/2)= 3/5 is θ=2*0.64350…+4pin,θ=2pi-2*0.64350…+4pin

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sin(θ/2)= 3/5

sin (\frac{θ}{2}) = \frac{3}{5}

θ = 2 \cdot 0.64350 \dots + 4 πn, θ = 2 π - 2 \cdot 0.64350 \dots + 4 πn

Degrees

θ = 73.73979 \dots^{\circ} + 72 0^{\circ} n, θ = 286.26020 \dots^{\circ} + 72 0^{\circ} n

Solution steps

sin (\frac{θ}{2}) = \frac{3}{5}

Apply trig inverse properties

sin (\frac{θ}{2}) = \frac{3}{5}

General solutions for

sin (\frac{θ}{2}) = \frac{3}{5}

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

\frac{θ}{2} = arcsin (\frac{3}{5}) + 2 πn, \frac{θ}{2} = π - arcsin (\frac{3}{5}) + 2 πn

\frac{θ}{2} = arcsin (\frac{3}{5}) + 2 πn, \frac{θ}{2} = π - arcsin (\frac{3}{5}) + 2 πn

Solve

\frac{θ}{2} = arcsin (\frac{3}{5}) + 2 πn : θ = 2 arcsin (\frac{3}{5}) + 4 πn

\frac{θ}{2} = arcsin (\frac{3}{5}) + 2 πn

Multiply both sides by

2

\frac{θ}{2} = arcsin (\frac{3}{5}) + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn : 2 arcsin (\frac{3}{5}) + 4 πn

2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 arcsin (\frac{3}{5}) + 4 πn

Solve

\frac{θ}{2} = π - arcsin (\frac{3}{5}) + 2 πn : θ = 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

\frac{θ}{2} = π - arcsin (\frac{3}{5}) + 2 πn

Multiply both sides by

2

\frac{θ}{2} = π - arcsin (\frac{3}{5}) + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 π - 2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 π - 2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 π - 2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn : 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

2 π - 2 arcsin (\frac{3}{5}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

θ = 2 arcsin (\frac{3}{5}) + 4 πn, θ = 2 π - 2 arcsin (\frac{3}{5}) + 4 πn

Show solutions in decimal form

θ = 2 \cdot 0.64350 \dots + 4 πn, θ = 2 π - 2 \cdot 0.64350 \dots + 4 πn

2 sin^{3} (t) + sin^{2} (t) - 2 sin (t) - 1 = 0

sin (x) = \frac{1}{5}, cos (x) > 0

2 sin^{2} (2 θ) = - 1 - 3 sin (2 θ)

6 cos (x) = 3

5 cot (x) = cos (x)

What is the general solution for sin(θ/2)= 3/5 ?
The general solution for sin(θ/2)= 3/5 is θ=2*0.64350…+4pin,θ=2pi-2*0.64350…+4pin