Precalculus Examples

$\sec (\arctan (\frac{1}{5} x))$

Step 1

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

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

Step 2

Combine $\frac{1}{5}$ and $x$ .

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

Step 3

Simplify the expression.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Rewrite $\frac{25 + x^{2}}{25}$ as ${(\frac{1}{5})}^{2} (25 + x^{2})$ .

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Step 5.1

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

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

Step 5.2

Factor the perfect power $5^{2}$ out of $25$ .

$\sqrt{\frac{1^{2} (25 + x^{2})}{5^{2} \cdot 1}}$

Step 5.3

Rearrange the fraction $\frac{1^{2} (25 + x^{2})}{5^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{5})}^{2} (25 + x^{2})}$

Step 6

Pull terms out from under the radical.

$\frac{1}{5} \sqrt{25 + x^{2}}$

Step 7

Combine $\frac{1}{5}$ and $\sqrt{25 + x^{2}}$ .

$\frac{\sqrt{25 + x^{2}}}{5}$