Trigonometry Examples

$\sec (- \frac{π}{12})$

Step 1

Replace $\sec (- \frac{π}{12})$ with an equivalent expression $\frac{1}{\cos (- \frac{π}{12})}$ using the fundamental identities.

$\frac{1}{\cos (- \frac{π}{12})}$

Step 2

Use a sum or difference formula on the denominator.

Tap for more steps...

Step 2.1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $- \frac{π}{12}$ can be split into $\frac{π}{4} - \frac{π}{3}$ .

$\frac{1}{\cos (\frac{π}{4} - \frac{π}{3})}$

Step 2.2

Use the difference formula for cosine to simplify the expression. The formula states that $\cos (A - B) = \cos (A) \cos (B) + \sin (A) \sin (B)$ .

$\frac{1}{\cos (\frac{π}{4}) \cdot \cos (\frac{π}{3}) + \sin (\frac{π}{4}) \cdot \sin (\frac{π}{3})}$

Step 2.3

Remove parentheses.

$\frac{1}{\cos (\frac{π}{4}) \cdot \cos (\frac{π}{3}) + \sin (\frac{π}{4}) \cdot \sin (\frac{π}{3})}$

Step 2.4

Simplify each term.

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

Step 2.4.3

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

Tap for more steps...

Step 2.4.3.1

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

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

Step 2.4.3.2

Multiply $2$ by $2$ .

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

Step 2.4.4

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

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

Step 2.4.5

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 2.4.6

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 2.4.6.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

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

Step 2.4.6.2

Combine using the product rule for radicals.

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

Step 2.4.6.3

Multiply $2$ by $3$ .

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

Step 2.4.6.4

Multiply $2$ by $2$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

Multiply $\frac{4}{\sqrt{2} + \sqrt{6}}$ by $1$ .

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

Step 3.4

Multiply $\frac{4}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

$\frac{4}{\sqrt{2} + \sqrt{6}} \cdot \frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$

Step 3.5

Multiply $\frac{4}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

$\frac{4 (\sqrt{2} - \sqrt{6})}{(\sqrt{2} + \sqrt{6}) (\sqrt{2} - \sqrt{6})}$

Step 3.6

Expand the denominator using the FOIL method.

$\frac{4 (\sqrt{2} - \sqrt{6})}{{\sqrt{2}}^{2} - \sqrt{12} + \sqrt{12} - {\sqrt{6}}^{2}}$

Step 3.7

Simplify.

$\frac{4 (\sqrt{2} - \sqrt{6})}{- 4}$

Step 3.8

Simplify the expression.

Tap for more steps...

Step 3.8.1

Move the negative one from the denominator of $\frac{\sqrt{2} - \sqrt{6}}{- 1}$ .

$- 1 \cdot (\sqrt{2} - \sqrt{6})$

Step 3.8.2

Rewrite $- 1 \cdot (\sqrt{2} - \sqrt{6})$ as $- (\sqrt{2} - \sqrt{6})$ .

$- (\sqrt{2} - \sqrt{6})$

Step 3.9

Apply the distributive property.

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

Step 3.10

Multiply $- - \sqrt{6}$ .

Tap for more steps...

Step 3.10.1

Multiply $- 1$ by $- 1$ .

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

Step 3.10.2

Multiply $\sqrt{6}$ by $1$ .

$- \sqrt{2} + \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2} + \sqrt{6}$

Decimal Form:

$1.03527618 \dots$