Trigonometry Examples

\arcsin (\sin (\frac{11 π}{8}))

$\arcsin (\sin (\frac{11 π}{8}))$

Step 1

Rewrite

\frac{11 π}{8}

$\frac{11 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by

2

$2$ .

\arcsin (\sin (\frac{\frac{11 π}{4}}{2}))

$\arcsin (\sin (\frac{\frac{11 π}{4}}{2}))$

Step 2

Apply the sine half-angle identity.

\arcsin (\pm \sqrt{\frac{1 - \cos (\frac{11 π}{4})}{2}})

$\arcsin (\pm \sqrt{\frac{1 - \cos (\frac{11 π}{4})}{2}})$

Step 3

Change the

\pm

$\pm$ to

-

$-$ because sine is negative in the third quadrant.

\arcsin (- \sqrt{\frac{1 - \cos (\frac{11 π}{4})}{2}})

$\arcsin (- \sqrt{\frac{1 - \cos (\frac{11 π}{4})}{2}})$

Step 4

Simplify

- \sqrt{\frac{1 - \cos (\frac{11 π}{4})}{2}}

$- \sqrt{\frac{1 - \cos (\frac{11 π}{4})}{2}}$ .

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Step 4.1

Subtract full rotations of

2 π

$2 π$ until the angle is greater than or equal to

0

$0$ and less than

2 π

$2 π$ .

\arcsin (- \sqrt{\frac{1 - \cos (\frac{3 π}{4})}{2}})

$\arcsin (- \sqrt{\frac{1 - \cos (\frac{3 π}{4})}{2}})$

Step 4.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

\arcsin (- \sqrt{\frac{1 - - \cos (\frac{π}{4})}{2}})

$\arcsin (- \sqrt{\frac{1 - - \cos (\frac{π}{4})}{2}})$

Step 4.3

The exact value of

\cos (\frac{π}{4})

$\cos (\frac{π}{4})$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\arcsin (- \sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{2}})

$\arcsin (- \sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{2}})$

Step 4.4

Multiply

- - \frac{\sqrt{2}}{2}

$- - \frac{\sqrt{2}}{2}$ .

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Step 4.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\arcsin (- \sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{2}})

$\arcsin (- \sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{2}})$

Step 4.4.2

Multiply

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ by

1

$1$ .

\arcsin (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})

$\arcsin (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})$

\arcsin (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})

$\arcsin (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}})$

Step 4.5

Write

1

$1$ as a fraction with a common denominator.

\arcsin (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}})

$\arcsin (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}})$

Step 4.6

Combine the numerators over the common denominator.

\arcsin (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}})

$\arcsin (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}})$

Step 4.7

Multiply the numerator by the reciprocal of the denominator.

\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}})

$\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}})$

Step 4.8

Multiply

\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}

$\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 4.8.1

Multiply

\frac{2 + \sqrt{2}}{2}

$\frac{2 + \sqrt{2}}{2}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}})

$\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}})$

Step 4.8.2

Multiply

2

$2$ by

2

$2$ .

\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{4}})

$\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{4}})$

\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{4}})

$\arcsin (- \sqrt{\frac{2 + \sqrt{2}}{4}})$

Step 4.9

Rewrite

\sqrt{\frac{2 + \sqrt{2}}{4}}

$\sqrt{\frac{2 + \sqrt{2}}{4}}$ as

\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}

$\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})

$\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})$

Step 4.10

Simplify the denominator.

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Step 4.10.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}})

$\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}})$

Step 4.10.2

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

\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})

$\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})$

\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})

$\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})$

\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})

$\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})$

Step 5

The result can be shown in multiple forms.

Exact Form:

\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})

$\arcsin (- \frac{\sqrt{2 + \sqrt{2}}}{2})$

Decimal Form:

- 1.17809724 \dots

$- 1.17809724 \dots$