Trigonometry Examples

$\sec (\arctan (\frac{8}{9}))$

Step 1

Draw a triangle in the plane with vertices $(1, \frac{8}{9})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{8}{9})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{8}{9})$ . Therefore, $\sec (\arctan (\frac{8}{9}))$ is $\sqrt{1 + {(\frac{8}{9})}^{2}}$ .

$\sqrt{1 + {(\frac{8}{9})}^{2}}$

Step 2

Apply the product rule to $\frac{8}{9}$ .

$\sqrt{1 + \frac{8^{2}}{9^{2}}}$

Step 3

Raise $8$ to the power of $2$ .

$\sqrt{1 + \frac{64}{9^{2}}}$

Step 4

Raise $9$ to the power of $2$ .

$\sqrt{1 + \frac{64}{81}}$

Step 5

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

$\sqrt{\frac{81}{81} + \frac{64}{81}}$

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{81 + 64}{81}}$

Step 7

Add $81$ and $64$ .

$\sqrt{\frac{145}{81}}$

Step 8

Rewrite $\sqrt{\frac{145}{81}}$ as $\frac{\sqrt{145}}{\sqrt{81}}$ .

$\frac{\sqrt{145}}{\sqrt{81}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $81$ as $9^{2}$ .

$\frac{\sqrt{145}}{\sqrt{9^{2}}}$

Step 9.2

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

$\frac{\sqrt{145}}{9}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{145}}{9}$

Decimal Form:

$1.33795495 \dots$