Algebra Examples

$\cos (\frac{29 π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{29 π}{12}$ can be split into $\frac{9 π}{4} + \frac{π}{6}$ .

$\cos (\frac{9 π}{4} + \frac{π}{6})$

Step 2

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

$\cos (\frac{π}{6}) \cdot \cos (\frac{9 π}{4}) - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 3

Remove parentheses.

$\cos (\frac{π}{6}) \cdot \cos (\frac{9 π}{4}) - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} \cdot \cos (\frac{9 π}{4}) - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\sqrt{3}}{2} \cdot \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.3

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

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.4.2

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.4.3

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2 \cdot 2} - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.4.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \sin (\frac{π}{6}) \cdot \sin (\frac{9 π}{4})$

Step 4.5

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \sin (\frac{9 π}{4})$

Step 4.6

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \sin (\frac{π}{4})$

Step 4.7

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

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

Step 4.8

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

Tap for more steps...

Step 4.8.1

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

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

Step 4.8.2

Multiply $2$ by $2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$