Trigonometry Examples

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

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

Step 1

Take the inverse arcsine of both sides of the equation to extract

x

$x$ from inside the arcsine.

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

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

Step 2

Simplify the right side.

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Step 2.1

Simplify

sin (arccos (\frac{12}{13}))

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

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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}} ⎞ ⎠

$(\frac{12}{13}, \sqrt{1^{2} - {(\frac{12}{13})}^{2}})$ ,

(\frac{12}{13}, 0)

$(\frac{12}{13}, 0)$ , and the origin. Then

arccos (\frac{12}{13})

$\arccos (\frac{12}{13})$ 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}} ⎞ ⎠

$(\frac{12}{13}, \sqrt{1^{2} - {(\frac{12}{13})}^{2}})$ . Therefore,

sin (arccos (\frac{12}{13}))

$\sin (\arccos (\frac{12}{13}))$ is

\sqrt{\frac{25}{169}}

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

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

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

Step 2.1.2

Rewrite

\sqrt{\frac{25}{169}}

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

\frac{\sqrt{25}}{\sqrt{169}}

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

x = \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

$25$ as

5^{2}

$5^{2}$ .

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

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

Step 2.1.3.2

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

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

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

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

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

Step 2.1.4

Simplify the denominator.

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Step 2.1.4.1

Rewrite

169

$169$ as

13^{2}

$13^{2}$ .

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

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

x = \frac{5}{13}

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

x = \frac{5}{13}

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

x = \frac{5}{13}

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

Step 3

The result can be shown in multiple forms.

Exact Form:

x = \frac{5}{13}

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

Decimal Form:

x = 0. ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ 384615

$x = 0.\overline{384615}$