Trigonometry Examples

$\sec (\arctan (\frac{6}{7}))$

Step 1

Draw a triangle in the plane with vertices $(1, \frac{6}{7})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{6}{7})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{6}{7})$ . Therefore, $\sec (\arctan (\frac{6}{7}))$ is $\sqrt{1 + {(\frac{6}{7})}^{2}}$ .

$\sqrt{1 + {(\frac{6}{7})}^{2}}$

Step 2

Apply the product rule to $\frac{6}{7}$ .

$\sqrt{1 + \frac{6^{2}}{7^{2}}}$

Step 3

Raise $6$ to the power of $2$ .

$\sqrt{1 + \frac{36}{7^{2}}}$

Step 4

Raise $7$ to the power of $2$ .

$\sqrt{1 + \frac{36}{49}}$

Step 5

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

$\sqrt{\frac{49}{49} + \frac{36}{49}}$

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{49 + 36}{49}}$

Step 7

Add $49$ and $36$ .

$\sqrt{\frac{85}{49}}$

Step 8

Rewrite $\sqrt{\frac{85}{49}}$ as $\frac{\sqrt{85}}{\sqrt{49}}$ .

$\frac{\sqrt{85}}{\sqrt{49}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{85}}{\sqrt{7^{2}}}$

Step 9.2

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

$\frac{\sqrt{85}}{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{85}}{7}$

Decimal Form:

$1.31707777 \dots$