Trigonometry Examples

sin (\frac{x}{2}) = \sqrt{\frac{1 - cos (x)}{2}}

$\sin (\frac{x}{2}) = \sqrt{\frac{1 - \cos (x)}{2}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

\sqrt{\frac{1 - cos (x)}{2}} = sin (\frac{x}{2})

$\sqrt{\frac{1 - \cos (x)}{2}} = \sin (\frac{x}{2})$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

{\sqrt{\frac{1 - cos (x)}{2}}}^{2} = {sin}^{2} (\frac{x}{2})

${\sqrt{\frac{1 - \cos (x)}{2}}}^{2} = \sin^{2} (\frac{x}{2})$

Step 3

Simplify each side of the equation.

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{\frac{1 - cos (x)}{2}}

$\sqrt{\frac{1 - \cos (x)}{2}}$ as

{(\frac{1 - cos (x)}{2})}^{\frac{1}{2}}

${(\frac{1 - \cos (x)}{2})}^{\frac{1}{2}}$ .

{({(\frac{1 - cos (x)}{2})}^{\frac{1}{2}})}^{2} = {sin}^{2} (\frac{x}{2})

${({(\frac{1 - \cos (x)}{2})}^{\frac{1}{2}})}^{2} = \sin^{2} (\frac{x}{2})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify

{({(\frac{1 - cos (x)}{2})}^{\frac{1}{2}})}^{2}

${({(\frac{1 - \cos (x)}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in

{({(\frac{1 - cos (x)}{2})}^{\frac{1}{2}})}^{2}

${({(\frac{1 - \cos (x)}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

{(\frac{1 - cos (x)}{2})}^{\frac{1}{2} \cdot 2} = {sin}^{2} (\frac{x}{2})

${(\frac{1 - \cos (x)}{2})}^{\frac{1}{2} \cdot 2} = \sin^{2} (\frac{x}{2})$

Step 3.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{1 - \cos (x)}{2})}^{\frac{1}{2} \cdot 2} = \sin^{2} (\frac{x}{2})$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{1 - \cos (x)}{2})}^{1} = \sin^{2} (\frac{x}{2})$

Step 3.2.1.2

Simplify.

$\frac{1 - \cos (x)}{2} = \sin^{2} (\frac{x}{2})$

Step 4

Solve for

$x$ .

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Step 4.1

Subtract

$\sin^{2} (\frac{x}{2})$ from both sides of the equation.

$\frac{1 - \cos (x)}{2} - \sin^{2} (\frac{x}{2}) = 0$

Step 4.2

Simplify

$\frac{1 - \cos (x)}{2} - \sin^{2} (\frac{x}{2})$ .

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Step 4.2.1

To write

$- \sin^{2} (\frac{x}{2})$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{1 - \cos (x)}{2} - \sin^{2} (\frac{x}{2}) \cdot \frac{2}{2} = 0$

Step 4.2.2

Combine

$- \sin^{2} (\frac{x}{2})$ and

$\frac{2}{2}$ .

$\frac{1 - \cos (x)}{2} + \frac{- \sin^{2} (\frac{x}{2}) \cdot 2}{2} = 0$

Step 4.2.3

Combine the numerators over the common denominator.

$\frac{1 - \cos (x) - \sin^{2} (\frac{x}{2}) \cdot 2}{2} = 0$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Multiply

$2$ by

$- 1$ .

$\frac{1 - \cos (x) - 2 \sin^{2} (\frac{x}{2})}{2} = 0$

Step 4.2.4.2

Move

$- 2 \sin^{2} (\frac{x}{2})$ .

$\frac{1 - 2 \sin^{2} (\frac{x}{2}) - \cos (x)}{2} = 0$

Step 4.2.4.3

Apply the cosine double-angle identity.

$\frac{\cos (2 \frac{x}{2}) - \cos (x)}{2} = 0$

Step 4.2.4.4

Cancel the common factor of

$2$ .

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Step 4.2.4.4.1

Cancel the common factor.

$\frac{\cos (2 \frac{x}{2}) - \cos (x)}{2} = 0$

Step 4.2.4.4.2

Rewrite the expression.

$\frac{\cos (x) - \cos (x)}{2} = 0$

Step 4.2.4.5

Subtract

$\cos (x)$ from

$\cos (x)$ .

$\frac{0}{2} = 0$

Step 4.2.5

Divide

$0$ by

$2$ .

$0 = 0$

Step 4.3

Since

$0 = 0$ , the equation will always be true for any value of

$x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$