Trigonometry Examples

$\arcsin (\sin (\frac{11 π}{8}))$

Step 1

Rewrite $\frac{11 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

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

Step 2

Apply the sine half-angle identity.

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

Step 3

Change the $\pm$ to $-$ because sine is negative in the third quadrant.

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

Step 4

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

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Step 4.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

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.

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

Step 4.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.4

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

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.4.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

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

Step 4.5

Write $1$ as a fraction with a common denominator.

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.8

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

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Step 4.8.1

Multiply $\frac{2 + \sqrt{2}}{2}$ by $\frac{1}{2}$ .

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

Step 4.8.2

Multiply $2$ by $2$ .

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

Step 4.9

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

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

Step 4.10

Simplify the denominator.

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Step 4.10.1

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

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

Step 4.10.2

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.17809724 \dots$