Trigonometry Examples

$\frac{3}{\tan (67.5)}$

Step 1

Separate fractions.

$\frac{3}{1} \cdot \frac{1}{\tan (67.5)}$

Step 2

Rewrite $\tan (67.5)$ in terms of sines and cosines.

$\frac{3}{1} \cdot \frac{1}{\frac{\sin (67.5)}{\cos (67.5)}}$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (67.5)}{\cos (67.5)}$ .

$\frac{3}{1} \cdot \frac{\cos (67.5)}{\sin (67.5)}$

Step 4

Convert from $\frac{\cos (67.5)}{\sin (67.5)}$ to $\cot (67.5)$ .

$\frac{3}{1} \cot (67.5)$

Step 5

Divide $3$ by $1$ .

$3 \cot (67.5)$

Step 6

The exact value of $\cot (67.5)$ is $\frac{1}{\sqrt{3 + 2 \sqrt{2}}}$ .

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Step 6.1

Rewrite $67.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$3 \cot (\frac{135}{2})$

Step 6.2

Apply the reciprocal identity.

$3 \frac{1}{\tan (\frac{135}{2})}$

Step 6.3

Apply the tangent half-angle identity.

$3 \frac{1}{\pm \sqrt{\frac{1 - \cos (135)}{1 + \cos (135)}}}$

Step 6.4

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

$3 \frac{1}{\sqrt{\frac{1 - \cos (135)}{1 + \cos (135)}}}$

Step 6.5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (135)}{1 + \cos (135)}}}$ .

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

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.

$3 \frac{1}{\sqrt{\frac{1 - - \cos (45)}{1 + \cos (135)}}}$

Step 6.5.1.2

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

$3 \frac{1}{\sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{1 + \cos (135)}}}$

Step 6.5.1.3

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

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Step 6.5.1.3.1

Multiply $- 1$ by $- 1$ .

$3 \frac{1}{\sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{1 + \cos (135)}}}$

Step 6.5.1.3.2

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

$3 \frac{1}{\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 + \cos (135)}}}$

Step 6.5.1.4

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

$3 \frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{1 + \cos (135)}}}$

Step 6.5.1.5

Combine the numerators over the common denominator.

$3 \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 + \cos (135)}}}$

Step 6.5.2

Simplify the denominator.

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Step 6.5.2.1

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.

$3 \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 - \cos (45)}}}$

Step 6.5.2.2

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

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

Step 6.5.2.3

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

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

Step 6.5.2.4

Combine the numerators over the common denominator.

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

Step 6.5.3

Simplify the denominator.

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Step 6.5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.5.3.2

Cancel the common factor of $2$ .

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Step 6.5.3.2.1

Cancel the common factor.

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

Step 6.5.3.2.2

Rewrite the expression.

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

Step 6.5.3.3

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

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

Step 6.5.3.4

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

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

Step 6.5.3.5

Expand the denominator using the FOIL method.

$3 \frac{1}{\sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}}}$

Step 6.5.3.6

Simplify.

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

Step 6.5.3.7

Apply the distributive property.

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

Step 6.5.3.8

Cancel the common factor of $2$ .

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Step 6.5.3.8.1

Cancel the common factor.

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

Step 6.5.3.8.2

Rewrite the expression.

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

Step 6.5.3.9

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

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

Step 6.5.3.10

Find the common denominator.

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Step 6.5.3.10.1

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

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

Step 6.5.3.10.2

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

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

Step 6.5.3.10.3

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

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

Step 6.5.3.10.4

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

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

Step 6.5.3.10.5

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

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

Step 6.5.3.10.6

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

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

Step 6.5.3.11

Combine the numerators over the common denominator.

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

Step 6.5.3.12

Simplify each term.

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Step 6.5.3.12.1

Multiply $2$ by $2$ .

$3 \frac{1}{\sqrt{\frac{4 + \sqrt{2} \cdot 2 + \sqrt{2} (2 + \sqrt{2})}{2}}}$

Step 6.5.3.12.2

Move $2$ to the left of $\sqrt{2}$ .

$3 \frac{1}{\sqrt{\frac{4 + 2 \cdot \sqrt{2} + \sqrt{2} (2 + \sqrt{2})}{2}}}$

Step 6.5.3.12.3

Apply the distributive property.

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

Step 6.5.3.12.4

Move $2$ to the left of $\sqrt{2}$ .

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

Step 6.5.3.12.5

Combine using the product rule for radicals.

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

Step 6.5.3.12.6

Simplify each term.

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Step 6.5.3.12.6.1

Multiply $2$ by $2$ .

$3 \frac{1}{\sqrt{\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{4}}{2}}}$

Step 6.5.3.12.6.2

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

$3 \frac{1}{\sqrt{\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{2^{2}}}{2}}}$

Step 6.5.3.12.6.3

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

$3 \frac{1}{\sqrt{\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + 2}{2}}}$

Step 6.5.3.13

Add $4$ and $2$ .

$3 \frac{1}{\sqrt{\frac{6 + 2 \sqrt{2} + 2 \sqrt{2}}{2}}}$

Step 6.5.3.14

Add $2 \sqrt{2}$ and $2 \sqrt{2}$ .

$3 \frac{1}{\sqrt{\frac{6 + 4 \sqrt{2}}{2}}}$

Step 6.5.3.15

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

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Step 6.5.3.15.1

Factor $2$ out of $6$ .

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

Step 6.5.3.15.2

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

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

Step 6.5.3.15.3

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

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

Step 6.5.3.15.4

Cancel the common factors.

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Step 6.5.3.15.4.1

Factor $2$ out of $2$ .

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

Step 6.5.3.15.4.2

Cancel the common factor.

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

Step 6.5.3.15.4.3

Rewrite the expression.

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

Step 6.5.3.15.4.4

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

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

Step 7

Combine $3$ and $\frac{1}{\sqrt{3 + 2 \sqrt{2}}}$ .

$\frac{3}{\sqrt{3 + 2 \sqrt{2}}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{\sqrt{3 + 2 \sqrt{2}}}$

Decimal Form:

$1.24264068 \dots$