What is the general solution for sin(θ/2)= 1/m ?

The general solution for sin(θ/2)= 1/m is θ=2arcsin(1/m)+4pin,θ=2pi+2arcsin(-1/m)+4pin

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sin(θ/2)= 1/m

sin (\frac{θ}{2}) = \frac{1}{m}

θ = 2 arcsin (\frac{1}{m}) + 4 πn, θ = 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

Solution steps

sin (\frac{θ}{2}) = \frac{1}{m}

Apply trig inverse properties

sin (\frac{θ}{2}) = \frac{1}{m}

General solutions for

sin (\frac{θ}{2}) = \frac{1}{m}

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

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

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

Solve

\frac{θ}{2} = arcsin (\frac{1}{m}) + 2 πn : θ = 2 arcsin (\frac{1}{m}) + 4 πn

\frac{θ}{2} = arcsin (\frac{1}{m}) + 2 πn

Multiply both sides by

2

\frac{θ}{2} = arcsin (\frac{1}{m}) + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 arcsin (\frac{1}{m}) + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 arcsin (\frac{1}{m}) + 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{1}{m}) + 2 \cdot 2 πn : 2 arcsin (\frac{1}{m}) + 4 πn

2 arcsin (\frac{1}{m}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arcsin (\frac{1}{m}) + 4 πn

θ = 2 arcsin (\frac{1}{m}) + 4 πn

θ = 2 arcsin (\frac{1}{m}) + 4 πn

θ = 2 arcsin (\frac{1}{m}) + 4 πn

Solve

\frac{θ}{2} = π + arcsin (- \frac{1}{m}) + 2 πn : θ = 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 θ}{2} = 2 π + 2 arcsin (- \frac{1}{m}) + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 π + 2 arcsin (- \frac{1}{m}) + 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{1}{m}) + 2 \cdot 2 πn : 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

2 π + 2 arcsin (- \frac{1}{m}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

θ = 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

θ = 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

θ = 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

θ = 2 arcsin (\frac{1}{m}) + 4 πn, θ = 2 π + 2 arcsin (- \frac{1}{m}) + 4 πn

2 (cos^{4} (x) + 1 - sin^{4} (x)) = 3

csc (\frac{2 x}{3}) - 1 = 0

6 tan (x) = tan^{2} (x), x = 0

5 cos^{2} (x) + 3 cos (x) - 1 = 0

3 cot (x) + 5 = 8

What is the general solution for sin(θ/2)= 1/m ?
The general solution for sin(θ/2)= 1/m is θ=2arcsin(1/m)+4pin,θ=2pi+2arcsin(-1/m)+4pin