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solvefor c,s=(sin^2(c))/2

Solution

solve for

c, s = \frac{sin ^{2} ( c )}{2}

c = arcsin (2 s) + 2 πn, c = π + arcsin (- 2 s) + 2 πn, c = arcsin (- 2 s) + 2 πn, c = π + arcsin (2 s) + 2 πn

Solution steps

s = \frac{sin ^{2} ( c )}{2}

Switch sides

\frac{sin ^{2} ( c )}{2} = s

Solve by substitution

\frac{sin ^{2} ( c )}{2} = s

Let:

sin (c) = u

\frac{u ^{2}}{2} = s

\frac{u ^{2}}{2} = s : u = 2 s, u = - 2 s

\frac{u ^{2}}{2} = s

Multiply both sides by

2

\frac{u ^{2}}{2} = s

Multiply both sides by

2

\frac{2 u ^{2}}{2} = 2 s

Simplify

u^{2} = 2 s

u^{2} = 2 s

For

x^{2} = f (a)

the solutions are

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

u = 2 s, u = - 2 s

Simplify

2 s : 2 s

2 s

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 2 s

Simplify

- 2 s : - 2 s

- 2 s

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= - 2 s

u = 2 s, u = - 2 s

Substitute back

u = sin (c)

sin (c) = 2 s, sin (c) = - 2 s

sin (c) = 2 s, sin (c) = - 2 s

sin (c) = 2 s : c = arcsin (2 s) + 2 πn, c = π + arcsin (- 2 s) + 2 πn

sin (c) = 2 s

Apply trig inverse properties

sin (c) = 2 s

General solutions for

sin (c) = 2 s

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

c = arcsin (2 s) + 2 πn, c = π + arcsin (- 2 s) + 2 πn

c = arcsin (2 s) + 2 πn, c = π + arcsin (- 2 s) + 2 πn

sin (c) = - 2 s : c = arcsin (- 2 s) + 2 πn, c = π + arcsin (2 s) + 2 πn

sin (c) = - 2 s

Apply trig inverse properties

sin (c) = - 2 s

General solutions for

sin (c) = - 2 s

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

c = arcsin (- 2 s) + 2 πn, c = π + arcsin (2 s) + 2 πn

c = arcsin (- 2 s) + 2 πn, c = π + arcsin (2 s) + 2 πn

Combine all the solutions

c = arcsin (2 s) + 2 πn, c = π + arcsin (- 2 s) + 2 πn, c = arcsin (- 2 s) + 2 πn, c = π + arcsin (2 s) + 2 πn

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