Trigonometry Examples

$\tan (arcsec (\frac{7}{4}))$

Step 1

Draw a triangle in the plane with vertices $(1, \sqrt{{(\frac{7}{4})}^{2} - 1^{2}})$ , $(1, 0)$ , and the origin. Then $arcsec (\frac{7}{4})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \sqrt{{(\frac{7}{4})}^{2} - 1^{2}})$ . Therefore, $\tan (arcsec (\frac{7}{4}))$ is $\sqrt{{(\frac{7}{4})}^{2} - 1}$ .

$\sqrt{{(\frac{7}{4})}^{2} - 1}$

Step 2

Apply the product rule to $\frac{7}{4}$ .

$\sqrt{\frac{7^{2}}{4^{2}} - 1}$

Step 3

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

$\sqrt{\frac{49}{4^{2}} - 1}$

Step 4

Raise $4$ to the power of $2$ .

$\sqrt{\frac{49}{16} - 1}$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{\frac{49}{16} - 1 \cdot \frac{16}{16}}$

Step 6

Combine $- 1$ and $\frac{16}{16}$ .

$\sqrt{\frac{49}{16} + \frac{- 1 \cdot 16}{16}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{49 - 1 \cdot 16}{16}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $16$ .

$\sqrt{\frac{49 - 16}{16}}$

Step 8.2

Subtract $16$ from $49$ .

$\sqrt{\frac{33}{16}}$

Step 9

Rewrite $\sqrt{\frac{33}{16}}$ as $\frac{\sqrt{33}}{\sqrt{16}}$ .

$\frac{\sqrt{33}}{\sqrt{16}}$

Step 10

Simplify the denominator.

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Step 10.1

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

$\frac{\sqrt{33}}{\sqrt{4^{2}}}$

Step 10.2

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

$\frac{\sqrt{33}}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{33}}{4}$

Decimal Form:

$1.43614066 \dots$