Trigonometry Examples

$\tan (67.5 °)$

Step 1

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

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

Step 2

Apply the tangent half-angle identity.

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

Step 3

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

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

Step 4

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

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Step 4.1

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.

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

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.

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Combine the numerators over the common denominator.

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

Step 4.10

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.11

Cancel the common factor of $2$ .

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Step 4.11.1

Cancel the common factor.

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

Step 4.11.2

Rewrite the expression.

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

Expand the denominator using the FOIL method.

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

Step 4.15

Simplify.

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

Step 4.16

Apply the distributive property.

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

Step 4.17

Cancel the common factor of $2$ .

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Step 4.17.1

Cancel the common factor.

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

Step 4.17.2

Rewrite the expression.

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

Step 4.18

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

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

Step 4.19

Simplify each term.

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Step 4.19.1

Apply the distributive property.

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

Step 4.19.2

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

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

Step 4.19.3

Combine using the product rule for radicals.

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

Step 4.19.4

Simplify each term.

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Step 4.19.4.1

Multiply $2$ by $2$ .

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

Step 4.19.4.2

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

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

Step 4.19.4.3

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

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

Step 4.19.5

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

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Step 4.19.5.1

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

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

Step 4.19.5.2

Factor $2$ out of $2$ .

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

Step 4.19.5.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (1)$ .

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

Step 4.19.5.4

Cancel the common factors.

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Step 4.19.5.4.1

Factor $2$ out of $2$ .

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

Step 4.19.5.4.2

Cancel the common factor.

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

Step 4.19.5.4.3

Rewrite the expression.

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

Step 4.19.5.4.4

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

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

Step 4.20

Add $2$ and $1$ .

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

Step 4.21

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.41421356 \dots$