Trigonometry Examples

$\csc (\frac{23 π}{8})$

Step 1

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

$\csc (\frac{7 π}{8})$

Step 2

The exact value of $\csc (\frac{7 π}{8})$ is $\frac{2}{\sqrt{2 - \sqrt{2}}}$ .

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Step 2.1

Rewrite $\frac{7 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\csc (\frac{\frac{7 π}{4}}{2})$

Step 2.2

Apply the reciprocal identity to $\csc (\frac{\frac{7 π}{4}}{2})$ .

$\frac{1}{\sin (\frac{\frac{7 π}{4}}{2})}$

Step 2.3

Apply the sine half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (\frac{7 π}{4})}{2}}}$

Step 2.4

Change the $\pm$ to $+$ because cosecant is positive in the second quadrant.

$\frac{1}{\sqrt{\frac{1 - \cos (\frac{7 π}{4})}{2}}}$

Step 2.5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (\frac{7 π}{4})}{2}}}$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

Combine the numerators over the common denominator.

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

Step 2.5.2

Simplify the denominator.

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Step 2.5.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.2.2

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

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Multiply $2$ by $2$ .

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

Step 2.5.2.3

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 2.5.2.4

Simplify the denominator.

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Step 2.5.2.4.1

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

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

Step 2.5.2.4.2

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

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

Step 2.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.4

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.61312592 \dots$