Trigonometry Examples

$\arcsin (x) = \arccos (\frac{12}{13})$

Step 1

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

$x = \sin (\arccos (\frac{12}{13}))$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\sin (\arccos (\frac{12}{13}))$ .

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Step 2.1.1

Draw a triangle in the plane with vertices $(\frac{12}{13}, \sqrt{1^{2} - {(\frac{12}{13})}^{2}})$ , $(\frac{12}{13}, 0)$ , and the origin. Then $\arccos (\frac{12}{13})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{12}{13}, \sqrt{1^{2} - {(\frac{12}{13})}^{2}})$ . Therefore, $\sin (\arccos (\frac{12}{13}))$ is $\sqrt{\frac{25}{169}}$ .

$x = \sqrt{\frac{25}{169}}$

Step 2.1.2

Rewrite $\sqrt{\frac{25}{169}}$ as $\frac{\sqrt{25}}{\sqrt{169}}$ .

$x = \frac{\sqrt{25}}{\sqrt{169}}$

Step 2.1.3

Simplify the numerator.

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Step 2.1.3.1

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

$x = \frac{\sqrt{5^{2}}}{\sqrt{169}}$

Step 2.1.3.2

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

$x = \frac{5}{\sqrt{169}}$

Step 2.1.4

Simplify the denominator.

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Step 2.1.4.1

Rewrite $169$ as $13^{2}$ .

$x = \frac{5}{\sqrt{13^{2}}}$

Step 2.1.4.2

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

$x = \frac{5}{13}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{13}$

Decimal Form:

$x = 0.\overline{384615}$