Trigonometry Examples

$\sec (\arctan (\frac{2}{5}))$

Step 1

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

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

Step 2

Apply the product rule to $\frac{2}{5}$ .

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

Step 3

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

$\sqrt{1 + \frac{4}{5^{2}}}$

Step 4

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

$\sqrt{1 + \frac{4}{25}}$

Step 5

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

$\sqrt{\frac{25}{25} + \frac{4}{25}}$

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{25 + 4}{25}}$

Step 7

Add $25$ and $4$ .

$\sqrt{\frac{29}{25}}$

Step 8

Rewrite $\sqrt{\frac{29}{25}}$ as $\frac{\sqrt{29}}{\sqrt{25}}$ .

$\frac{\sqrt{29}}{\sqrt{25}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{29}}{\sqrt{5^{2}}}$

Step 9.2

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

$\frac{\sqrt{29}}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{29}}{5}$

Decimal Form:

$1.07703296 \dots$