Trigonometry Examples

$\csc (\frac{11 π}{8})$

Step 1

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

$\csc (\frac{\frac{11 π}{4}}{2})$

Step 2

Apply the reciprocal identity to $\csc (\frac{\frac{11 π}{4}}{2})$ .

$\frac{1}{\sin (\frac{\frac{11 π}{4}}{2})}$

Step 3

Apply the sine half-angle identity.

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

Step 4

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

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

Step 5

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

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Step 5.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.1.2

Move the negative in front of the fraction.

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

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

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

Step 5.2.2

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.

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

Step 5.2.3

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

$- \frac{1}{\sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{2}}}$

Step 5.2.4

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

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Step 5.2.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{\sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{2}}}$

Step 5.2.4.2

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

$- \frac{1}{\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}}$

Step 5.2.5

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

$- \frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}}$

Step 5.2.6

Combine the numerators over the common denominator.

$- \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Multiply $2$ by $2$ .

$- \frac{1}{\sqrt{\frac{2 + \sqrt{2}}{4}}}$

Step 5.3.3

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

$- \frac{1}{\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

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

$- \frac{1}{\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}}}$

Step 5.3.4.2

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

$- \frac{1}{\frac{\sqrt{2 + \sqrt{2}}}{2}}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.08239220 \dots$