Trigonometry Examples

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

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.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{1 - \cos (a)}{2}}$ as ${(\frac{1 - \cos (a)}{2})}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $\frac{1 - \cos (a)}{2}$ .

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

Step 3.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.2.1

Apply the product rule to $- \frac{{(1 - \cos (a))}^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ .

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

Step 3.2.1.2.2

Apply the product rule to $\frac{{(1 - \cos (a))}^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ .

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

Step 3.2.1.3

Simplify the expression.

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Step 3.2.1.3.1

Raise $- 1$ to the power of $2$ .

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

Step 3.2.1.3.2

Multiply $\frac{{({(1 - \cos (a))}^{\frac{1}{2}})}^{2}}{{(2^{\frac{1}{2}})}^{2}}$ by $1$ .

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

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

Multiply the exponents in ${({(1 - \cos (a))}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.1.4.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.1.2.1

Cancel the common factor.

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

Step 3.2.1.4.1.2.2

Rewrite the expression.

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

Step 3.2.1.4.2

Simplify.

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

Step 3.2.1.5

Simplify the denominator.

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Step 3.2.1.5.1

Multiply the exponents in ${(2^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.1.5.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.1.2.1

Cancel the common factor.

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

Step 3.2.1.5.1.2.2

Rewrite the expression.

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

Step 3.2.1.5.2

Evaluate the exponent.

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

Step 4

Solve for $a$ .

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Combine $- \sin^{2} (\frac{a}{2})$ and $\frac{2}{2}$ .

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Multiply $2$ by $- 1$ .

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

Step 4.2.4.2

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

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

Step 4.2.4.3

Apply the cosine double-angle identity.

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

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.

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

Step 4.2.4.4.2

Rewrite the expression.

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

Step 4.2.4.5

Subtract $\cos (a)$ from $\cos (a)$ .

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

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 $a$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$