Precalculus Examples

$\cos (120 + 45)$

Step 1

Add $120$ and $45$ .

$\cos (165)$

Step 2

The exact value of $\cos (165)$ is $- \frac{\sqrt{6} + \sqrt{2}}{4}$ .

Tap for more steps...

Step 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.

$- \cos (15)$

Step 2.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$- \cos (45 - 30)$

Step 2.3

Separate negation.

$- \cos (45 - (30))$

Step 2.4

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$- (\cos (45) \cos (30) + \sin (45) \sin (30))$

Step 2.5

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

$- (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30))$

Step 2.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30))$

Step 2.7

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30))$

Step 2.8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Step 2.9

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

Tap for more steps...

Step 2.9.1

Simplify each term.

Tap for more steps...

Step 2.9.1.1

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

Tap for more steps...

Step 2.9.1.1.1

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

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

Step 2.9.1.1.2

Combine using the product rule for radicals.

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

Step 2.9.1.1.3

Multiply $2$ by $3$ .

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

Step 2.9.1.1.4

Multiply $2$ by $2$ .

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

Step 2.9.1.2

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

Tap for more steps...

Step 2.9.1.2.1

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

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

Step 2.9.1.2.2

Multiply $2$ by $2$ .

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

Step 2.9.2

Combine the numerators over the common denominator.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$