Trigonometry Examples

$\arccos (x) = \arcsin (\frac{15}{17})$

Step 1

Take the inverse arccosine of both sides of the equation to extract $x$ from inside the arccosine.

$x = \cos (\arcsin (\frac{15}{17}))$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\cos (\arcsin (\frac{15}{17}))$ .

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Step 2.1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{15}{17})}^{2}}, \frac{15}{17})$ , $(\sqrt{1^{2} - {(\frac{15}{17})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{15}{17})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{15}{17})}^{2}}, \frac{15}{17})$ . Therefore, $\cos (\arcsin (\frac{15}{17}))$ is $\sqrt{\frac{64}{289}}$ .

$x = \sqrt{\frac{64}{289}}$

Step 2.1.2

Rewrite $\sqrt{\frac{64}{289}}$ as $\frac{\sqrt{64}}{\sqrt{289}}$ .

$x = \frac{\sqrt{64}}{\sqrt{289}}$

Step 2.1.3

Simplify the numerator.

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Step 2.1.3.1

Rewrite $64$ as $8^{2}$ .

$x = \frac{\sqrt{8^{2}}}{\sqrt{289}}$

Step 2.1.3.2

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

$x = \frac{8}{\sqrt{289}}$

Step 2.1.4

Simplify the denominator.

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Step 2.1.4.1

Rewrite $289$ as $17^{2}$ .

$x = \frac{8}{\sqrt{17^{2}}}$

Step 2.1.4.2

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

$x = \frac{8}{17}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{17}$

Decimal Form:

$x = 0.47058823 \dots$