Precalculus Examples

$\sec (\arctan (\frac{5}{12}))$

Step 1

Draw a triangle in the plane with vertices $(1, \frac{5}{12})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{5}{12})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{5}{12})$ . Therefore, $\sec (\arctan (\frac{5}{12}))$ is $\sqrt{1 + {(\frac{5}{12})}^{2}}$ .

$\sqrt{1 + {(\frac{5}{12})}^{2}}$

Step 2

Apply the product rule to $\frac{5}{12}$ .

$\sqrt{1 + \frac{5^{2}}{12^{2}}}$

Step 3

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

$\sqrt{1 + \frac{25}{12^{2}}}$

Step 4

Raise $12$ to the power of $2$ .

$\sqrt{1 + \frac{25}{144}}$

Step 5

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

$\sqrt{\frac{144}{144} + \frac{25}{144}}$

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{144 + 25}{144}}$

Step 7

Add $144$ and $25$ .

$\sqrt{\frac{169}{144}}$

Step 8

Rewrite $\sqrt{\frac{169}{144}}$ as $\frac{\sqrt{169}}{\sqrt{144}}$ .

$\frac{\sqrt{169}}{\sqrt{144}}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite $169$ as $13^{2}$ .

$\frac{\sqrt{13^{2}}}{\sqrt{144}}$

Step 9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{13}{\sqrt{144}}$

Step 10

Simplify the denominator.

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Step 10.1

Rewrite $144$ as $12^{2}$ .

$\frac{13}{\sqrt{12^{2}}}$

Step 10.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{13}{12}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{12}$

Decimal Form:

$1.08\overline{3}$

Mixed Number Form:

$1 \frac{1}{12}$