Calculus Examples

$\sin (\arccos (\frac{3}{5}))$

Step 1

Draw a triangle in the plane with vertices $(\frac{3}{5}, \sqrt{1^{2} - {(\frac{3}{5})}^{2}})$ , $(\frac{3}{5}, 0)$ , and the origin. Then $\arccos (\frac{3}{5})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{3}{5}, \sqrt{1^{2} - {(\frac{3}{5})}^{2}})$ . Therefore, $\sin (\arccos (\frac{3}{5}))$ is $\sqrt{\frac{16}{25}}$ .

$\sqrt{\frac{16}{25}}$

Step 2

Rewrite $\sqrt{\frac{16}{25}}$ as $\frac{\sqrt{16}}{\sqrt{25}}$ .

$\frac{\sqrt{16}}{\sqrt{25}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{\sqrt{4^{2}}}{\sqrt{25}}$

Step 3.2

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

$\frac{4}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$