Trigonometry Examples

$\cos (\arctan (\frac{5}{x}))$

Step 1

Draw a triangle in the plane with vertices $(1, \frac{5}{x})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{5}{x})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{5}{x})$ . Therefore, $\cos (\arctan (\frac{5}{x}))$ is $\frac{1}{\sqrt{1 + {(\frac{5}{x})}^{2}}}$ .

$\frac{1}{\sqrt{1 + {(\frac{5}{x})}^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $\frac{5}{x}$ .

$\frac{1}{\sqrt{1 + \frac{5^{2}}{x^{2}}}}$

Step 2.2

Raise $5$ to the power of $2$ .

$\frac{1}{\sqrt{1 + \frac{25}{x^{2}}}}$

Step 2.3

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

$\frac{1}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{25}{x^{2}}}}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{x^{2} + 25}{x^{2}}}}$

Step 2.5

Rewrite $\frac{x^{2} + 25}{x^{2}}$ as ${(\frac{1}{x})}^{2} (x^{2} + 25)$ .

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Step 2.5.1

Factor the perfect power $1^{2}$ out of $x^{2} + 25$ .

$\frac{1}{\sqrt{\frac{1^{2} (x^{2} + 25)}{x^{2}}}}$

Step 2.5.2

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

$\frac{1}{\sqrt{\frac{1^{2} (x^{2} + 25)}{x^{2} \cdot 1}}}$

Step 2.5.3

Rearrange the fraction $\frac{1^{2} (x^{2} + 25)}{x^{2} \cdot 1}$ .

$\frac{1}{\sqrt{{(\frac{1}{x})}^{2} (x^{2} + 25)}}$

Step 2.6

Pull terms out from under the radical.

$\frac{1}{\frac{1}{x} \sqrt{x^{2} + 25}}$

Step 2.7

Combine $\frac{1}{x}$ and $\sqrt{x^{2} + 25}$ .

$\frac{1}{\frac{\sqrt{x^{2} + 25}}{x}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{x}{\sqrt{x^{2} + 25}}$

Step 4

Multiply $\frac{x}{\sqrt{x^{2} + 25}}$ by $1$ .

$\frac{x}{\sqrt{x^{2} + 25}}$

Step 5

Multiply $\frac{x}{\sqrt{x^{2} + 25}}$ by $\frac{\sqrt{x^{2} + 25}}{\sqrt{x^{2} + 25}}$ .

$\frac{x}{\sqrt{x^{2} + 25}} \cdot \frac{\sqrt{x^{2} + 25}}{\sqrt{x^{2} + 25}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{x}{\sqrt{x^{2} + 25}}$ by $\frac{\sqrt{x^{2} + 25}}{\sqrt{x^{2} + 25}}$ .

$\frac{x \sqrt{x^{2} + 25}}{\sqrt{x^{2} + 25} \sqrt{x^{2} + 25}}$

Step 6.2

Raise $\sqrt{x^{2} + 25}$ to the power of $1$ .

$\frac{x \sqrt{x^{2} + 25}}{{\sqrt{x^{2} + 25}}^{1} \sqrt{x^{2} + 25}}$

Step 6.3

Raise $\sqrt{x^{2} + 25}$ to the power of $1$ .

$\frac{x \sqrt{x^{2} + 25}}{{\sqrt{x^{2} + 25}}^{1} {\sqrt{x^{2} + 25}}^{1}}$

Step 6.4

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

$\frac{x \sqrt{x^{2} + 25}}{{\sqrt{x^{2} + 25}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{x \sqrt{x^{2} + 25}}{{\sqrt{x^{2} + 25}}^{2}}$

Step 6.6

Rewrite ${\sqrt{x^{2} + 25}}^{2}$ as $x^{2} + 25$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 25}$ as ${(x^{2} + 25)}^{\frac{1}{2}}$ .

$\frac{x \sqrt{x^{2} + 25}}{{({(x^{2} + 25)}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{x \sqrt{x^{2} + 25}}{{(x^{2} + 25)}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{x \sqrt{x^{2} + 25}}{{(x^{2} + 25)}^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{x \sqrt{x^{2} + 25}}{{(x^{2} + 25)}^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{x \sqrt{x^{2} + 25}}{{(x^{2} + 25)}^{1}}$

Step 6.6.5

Simplify.

$\frac{x \sqrt{x^{2} + 25}}{x^{2} + 25}$