Trigonometry Examples

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

Step 1

Draw a triangle in the plane with vertices $(1, \frac{x}{2})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{2})$ . Therefore, $\cos (\arctan (\frac{x}{2}))$ is $\frac{1}{\sqrt{1 + {(\frac{x}{2})}^{2}}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $\frac{x}{2}$ .

$\frac{1}{\sqrt{1 + \frac{x^{2}}{2^{2}}}}$

Step 2.2

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

$\frac{1}{\sqrt{1 + \frac{x^{2}}{4}}}$

Step 2.3

Write $1$ as a fraction with a common denominator.

$\frac{1}{\sqrt{\frac{4}{4} + \frac{x^{2}}{4}}}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{4 + x^{2}}{4}}}$

Step 2.5

Rewrite $\frac{4 + x^{2}}{4}$ as ${(\frac{1}{2})}^{2} (4 + x^{2})$ .

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Step 2.5.1

Factor the perfect power $1^{2}$ out of $4 + x^{2}$ .

$\frac{1}{\sqrt{\frac{1^{2} (4 + x^{2})}{4}}}$

Step 2.5.2

Factor the perfect power $2^{2}$ out of $4$ .

$\frac{1}{\sqrt{\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}}}$

Step 2.5.3

Rearrange the fraction $\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}$ .

$\frac{1}{\sqrt{{(\frac{1}{2})}^{2} (4 + x^{2})}}$

Step 2.6

Pull terms out from under the radical.

$\frac{1}{\frac{1}{2} \sqrt{4 + x^{2}}}$

Step 2.7

Combine $\frac{1}{2}$ and $\sqrt{4 + x^{2}}$ .

$\frac{1}{\frac{\sqrt{4 + x^{2}}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2}{\sqrt{4 + x^{2}}}$

Step 4

Multiply $\frac{2}{\sqrt{4 + x^{2}}}$ by $1$ .

$\frac{2}{\sqrt{4 + x^{2}}}$

Step 5

Multiply $\frac{2}{\sqrt{4 + x^{2}}}$ by $\frac{\sqrt{4 + x^{2}}}{\sqrt{4 + x^{2}}}$ .

$\frac{2}{\sqrt{4 + x^{2}}} \cdot \frac{\sqrt{4 + x^{2}}}{\sqrt{4 + x^{2}}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{2}{\sqrt{4 + x^{2}}}$ by $\frac{\sqrt{4 + x^{2}}}{\sqrt{4 + x^{2}}}$ .

$\frac{2 \sqrt{4 + x^{2}}}{\sqrt{4 + x^{2}} \sqrt{4 + x^{2}}}$

Step 6.2

Raise $\sqrt{4 + x^{2}}$ to the power of $1$ .

$\frac{2 \sqrt{4 + x^{2}}}{{\sqrt{4 + x^{2}}}^{1} \sqrt{4 + x^{2}}}$

Step 6.3

Raise $\sqrt{4 + x^{2}}$ to the power of $1$ .

$\frac{2 \sqrt{4 + x^{2}}}{{\sqrt{4 + x^{2}}}^{1} {\sqrt{4 + x^{2}}}^{1}}$

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{4 + x^{2}}}{{\sqrt{4 + x^{2}}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{2 \sqrt{4 + x^{2}}}{{\sqrt{4 + x^{2}}}^{2}}$

Step 6.6

Rewrite ${\sqrt{4 + x^{2}}}^{2}$ as $4 + x^{2}$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + x^{2}}$ as ${(4 + x^{2})}^{\frac{1}{2}}$ .

$\frac{2 \sqrt{4 + x^{2}}}{{({(4 + x^{2})}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{2 \sqrt{4 + x^{2}}}{{(4 + x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{4 + x^{2}}}{{(4 + x^{2})}^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{4 + x^{2}}}{{(4 + x^{2})}^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{4 + x^{2}}}{{(4 + x^{2})}^{1}}$

Step 6.6.5

Simplify.

$\frac{2 \sqrt{4 + x^{2}}}{4 + x^{2}}$