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

Tap for more steps...

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

Tap for more steps...

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 $2$ .

Tap for more steps...

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 $2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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.

Tap for more steps...

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:

$2.41421356 \dots$