Calculus Examples

$\tan (arcsec (\frac{5}{3}))$

Step 1

Draw a triangle in the plane with vertices $(1, \sqrt{{(\frac{5}{3})}^{2} - 1^{2}})$ , $(1, 0)$ , and the origin. Then $arcsec (\frac{5}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \sqrt{{(\frac{5}{3})}^{2} - 1^{2}})$ . Therefore, $\tan (arcsec (\frac{5}{3}))$ is $\sqrt{{(\frac{5}{3})}^{2} - 1}$ .

$\sqrt{{(\frac{5}{3})}^{2} - 1}$

Step 2

Apply the product rule to $\frac{5}{3}$ .

$\sqrt{\frac{5^{2}}{3^{2}} - 1}$

Step 3

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

$\sqrt{\frac{25}{3^{2}} - 1}$

Step 4

Raise $3$ to the power of $2$ .

$\sqrt{\frac{25}{9} - 1}$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{25}{9} - 1 \cdot \frac{9}{9}}$

Step 6

Combine $- 1$ and $\frac{9}{9}$ .

$\sqrt{\frac{25}{9} + \frac{- 1 \cdot 9}{9}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{25 - 1 \cdot 9}{9}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $9$ .

$\sqrt{\frac{25 - 9}{9}}$

Step 8.2

Subtract $9$ from $25$ .

$\sqrt{\frac{16}{9}}$

Step 9

Rewrite $\sqrt{\frac{16}{9}}$ as $\frac{\sqrt{16}}{\sqrt{9}}$ .

$\frac{\sqrt{16}}{\sqrt{9}}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}}}{\sqrt{9}}$

Step 10.2

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

$\frac{4}{\sqrt{9}}$

Step 11

Simplify the denominator.

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Step 11.1

Rewrite $9$ as $3^{2}$ .

$\frac{4}{\sqrt{3^{2}}}$

Step 11.2

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

$\frac{4}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$