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Popular >

sin^2(x/2)=a

Solution

sin^{2} (\frac{x}{2}) = a

Solution

x = 2 arcsin (a) + 4 πn, x = 2 π + 2 arcsin (- a) + 4 πn, x = 2 arcsin (- a) + 4 πn, x = 2 π + 2 arcsin (a) + 4 πn

Solution steps

sin^{2} (\frac{x}{2}) = a

Solve by substitution

sin^{2} (\frac{x}{2}) = a

Let:

sin (\frac{x}{2}) = u

u^{2} = a

u^{2} = a : u = a, u = - a

u^{2} = a

For

x^{2} = f (a)

the solutions are

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

u = a, u = - a

Substitute back

u = sin (\frac{x}{2})

sin (\frac{x}{2}) = a, sin (\frac{x}{2}) = - a

sin (\frac{x}{2}) = a, sin (\frac{x}{2}) = - a

sin (\frac{x}{2}) = a : x = 2 arcsin (a) + 4 πn, x = 2 π + 2 arcsin (- a) + 4 πn

sin (\frac{x}{2}) = a

Apply trig inverse properties

sin (\frac{x}{2}) = a

General solutions for

sin (\frac{x}{2}) = a

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

\frac{x}{2} = arcsin (a) + 2 πn, \frac{x}{2} = π + arcsin (- a) + 2 πn

\frac{x}{2} = arcsin (a) + 2 πn, \frac{x}{2} = π + arcsin (- a) + 2 πn

Solve

\frac{x}{2} = arcsin (a) + 2 πn : x = 2 arcsin (a) + 4 πn

\frac{x}{2} = arcsin (a) + 2 πn

Multiply both sides by

2

\frac{x}{2} = arcsin (a) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arcsin (a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 arcsin (a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arcsin (a) + 2 \cdot 2 πn : 2 arcsin (a) + 4 πn

2 arcsin (a) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arcsin (a) + 4 πn

x = 2 arcsin (a) + 4 πn

x = 2 arcsin (a) + 4 πn

x = 2 arcsin (a) + 4 πn

Solve

\frac{x}{2} = π + arcsin (- a) + 2 πn : x = 2 π + 2 arcsin (- a) + 4 πn

\frac{x}{2} = π + arcsin (- a) + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + arcsin (- a) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 arcsin (- a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 π + 2 arcsin (- a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 π + 2 arcsin (- a) + 2 \cdot 2 πn : 2 π + 2 arcsin (- a) + 4 πn

2 π + 2 arcsin (- a) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π + 2 arcsin (- a) + 4 πn

x = 2 π + 2 arcsin (- a) + 4 πn

x = 2 π + 2 arcsin (- a) + 4 πn

x = 2 π + 2 arcsin (- a) + 4 πn

x = 2 arcsin (a) + 4 πn, x = 2 π + 2 arcsin (- a) + 4 πn

sin (\frac{x}{2}) = - a : x = 2 arcsin (- a) + 4 πn, x = 2 π + 2 arcsin (a) + 4 πn

sin (\frac{x}{2}) = - a

Apply trig inverse properties

sin (\frac{x}{2}) = - a

General solutions for

sin (\frac{x}{2}) = - a

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

\frac{x}{2} = arcsin (- a) + 2 πn, \frac{x}{2} = π + arcsin (a) + 2 πn

\frac{x}{2} = arcsin (- a) + 2 πn, \frac{x}{2} = π + arcsin (a) + 2 πn

Solve

\frac{x}{2} = arcsin (- a) + 2 πn : x = 2 arcsin (- a) + 4 πn

\frac{x}{2} = arcsin (- a) + 2 πn

Multiply both sides by

2

\frac{x}{2} = arcsin (- a) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arcsin (- a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 arcsin (- a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arcsin (- a) + 2 \cdot 2 πn : 2 arcsin (- a) + 4 πn

2 arcsin (- a) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arcsin (- a) + 4 πn

x = 2 arcsin (- a) + 4 πn

x = 2 arcsin (- a) + 4 πn

x = 2 arcsin (- a) + 4 πn

Solve

\frac{x}{2} = π + arcsin (a) + 2 πn : x = 2 π + 2 arcsin (a) + 4 πn

\frac{x}{2} = π + arcsin (a) + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + arcsin (a) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 arcsin (a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 π + 2 arcsin (a) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 π + 2 arcsin (a) + 2 \cdot 2 πn : 2 π + 2 arcsin (a) + 4 πn

2 π + 2 arcsin (a) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π + 2 arcsin (a) + 4 πn

x = 2 π + 2 arcsin (a) + 4 πn

x = 2 π + 2 arcsin (a) + 4 πn

x = 2 π + 2 arcsin (a) + 4 πn

x = 2 arcsin (- a) + 4 πn, x = 2 π + 2 arcsin (a) + 4 πn

Combine all the solutions

x = 2 arcsin (a) + 4 πn, x = 2 π + 2 arcsin (- a) + 4 πn, x = 2 arcsin (- a) + 4 πn, x = 2 π + 2 arcsin (a) + 4 πn

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