Trigonometry Examples

cos (\frac{x}{2})

$\cos (\frac{x}{2})$

Step 1

Apply the cosine half-angle identity

cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + cos (x)}{2}}

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

\pm \sqrt{\frac{1 + cos (x)}{2}}

$\pm \sqrt{\frac{1 + \cos (x)}{2}}$

Step 2

Rewrite

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

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

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

$\frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$ .

\pm \frac{\sqrt{1 + cos (x)}}{\sqrt{2}}

$\pm \frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$

Step 3

Multiply

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

$\frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

\pm \frac{\sqrt{1 + cos (x)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}

$\pm \frac{\sqrt{1 + \cos (x)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply

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

$\frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{\sqrt{2} \sqrt{2}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.5

Add

1

$1$ and

1

$1$ .

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{{\sqrt{2}}^{2}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.6

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 4.6.1

Use

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

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

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.6.2

Apply the power rule and multiply exponents,

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

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

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\pm \frac{\sqrt{1 + cos (x)} \sqrt{2}}{2^{\frac{2}{2}}}

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of

2

$2$ .

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Step 4.6.4.1

Cancel the common factor.

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{2^{1}}$

Step 4.6.5

Evaluate the exponent.

$\pm \frac{\sqrt{1 + \cos (x)} \sqrt{2}}{2}$

Step 5

Combine using the product rule for radicals.

$\pm \frac{\sqrt{(1 + \cos (x)) \cdot 2}}{2}$

Step 6

Reorder factors in

$\pm \frac{\sqrt{(1 + \cos (x)) \cdot 2}}{2}$ .

$\pm \frac{\sqrt{2 (1 + \cos (x))}}{2}$