Trigonometry Examples

$\cot (22.5)$

Step 1

Rewrite $22.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cot (\frac{45}{2})$

Step 2

Apply the reciprocal identity.

$\frac{1}{\tan (\frac{45}{2})}$

Step 3

Apply the tangent half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}}$

Step 4

Change the $\pm$ to $+$ because cotangent is positive in the first quadrant.

$\frac{1}{\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}}$

Step 5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{1}{\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}}$

Step 5.1.2

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

$\frac{1}{\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}}$

Step 5.1.3

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \cos (45)}}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 5.2.2

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2

Cancel the common factor of $2$ .

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Step 5.3.2.1

Cancel the common factor.

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

Step 5.3.2.2

Rewrite the expression.

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.3.5

Expand the denominator using the FOIL method.

$\frac{1}{\sqrt{(2 - \sqrt{2}) \frac{2 - \sqrt{2}}{4 - 2 \sqrt{2} + \sqrt{2} \cdot 2 - {\sqrt{2}}^{2}}}}$

Step 5.3.6

Simplify.

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

Step 5.3.7

Apply the distributive property.

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

Step 5.3.8

Cancel the common factor of $2$ .

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Step 5.3.8.1

Cancel the common factor.

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

Step 5.3.8.2

Rewrite the expression.

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

Step 5.3.9

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

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

Step 5.3.10

Find the common denominator.

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Step 5.3.10.1

Write $2$ as a fraction with denominator $1$ .

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

Step 5.3.10.2

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

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

Step 5.3.10.3

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

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

Step 5.3.10.4

Write $- \sqrt{2}$ as a fraction with denominator $1$ .

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

Step 5.3.10.5

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

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

Step 5.3.10.6

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

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

Step 5.3.11

Combine the numerators over the common denominator.

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

Step 5.3.12

Simplify each term.

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Step 5.3.12.1

Multiply $2$ by $2$ .

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

Step 5.3.12.2

Multiply $2$ by $- 1$ .

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

Step 5.3.12.3

Apply the distributive property.

$\frac{1}{\sqrt{\frac{4 - 2 \sqrt{2} + (- 1 \cdot 2 - - \sqrt{2}) \sqrt{2}}{2}}}$

Step 5.3.12.4

Multiply $- 1$ by $2$ .

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

Step 5.3.12.5

Multiply $- - \sqrt{2}$ .

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Step 5.3.12.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.12.5.2

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

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

Step 5.3.12.6

Apply the distributive property.

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

Step 5.3.12.7

Combine using the product rule for radicals.

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

Step 5.3.12.8

Simplify each term.

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Step 5.3.12.8.1

Multiply $2$ by $2$ .

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

Step 5.3.12.8.2

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

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

Step 5.3.12.8.3

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

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

Step 5.3.13

Add $4$ and $2$ .

$\frac{1}{\sqrt{\frac{6 - 2 \sqrt{2} - 2 \sqrt{2}}{2}}}$

Step 5.3.14

Subtract $2 \sqrt{2}$ from $- 2 \sqrt{2}$ .

$\frac{1}{\sqrt{\frac{6 - 4 \sqrt{2}}{2}}}$

Step 5.3.15

Cancel the common factor of $6 - 4 \sqrt{2}$ and $2$ .

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Step 5.3.15.1

Factor $2$ out of $6$ .

$\frac{1}{\sqrt{\frac{2 \cdot 3 - 4 \sqrt{2}}{2}}}$

Step 5.3.15.2

Factor $2$ out of $- 4 \sqrt{2}$ .

$\frac{1}{\sqrt{\frac{2 \cdot 3 + 2 (- 2 \sqrt{2})}{2}}}$

Step 5.3.15.3

Factor $2$ out of $2 (3) + 2 (- 2 \sqrt{2})$ .

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

Step 5.3.15.4

Cancel the common factors.

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Step 5.3.15.4.1

Factor $2$ out of $2$ .

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

Step 5.3.15.4.2

Cancel the common factor.

$\frac{1}{\sqrt{\frac{2 (3 - 2 \sqrt{2})}{2 \cdot 1}}}$

Step 5.3.15.4.3

Rewrite the expression.

$\frac{1}{\sqrt{\frac{3 - 2 \sqrt{2}}{1}}}$

Step 5.3.15.4.4

Divide $3 - 2 \sqrt{2}$ by $1$ .

$\frac{1}{\sqrt{3 - 2 \sqrt{2}}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{\sqrt{3 - 2 \sqrt{2}}}$

Decimal Form:

$2.41421356 \dots$