Find the Exact Value cos(120 degrees +45 degrees )

$\cos (120 ° + 45 °)$

Step 1

Add $120 °$ and $45 °$ .

$\cos (165)$

Step 2

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

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.

$- \cos (15)$

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

$- \cos (45 - 30)$

Separate negation.

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

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

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

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:

$- 0.96592582 \dots$

Find the Exact Value cos(120 degrees +45 degrees )

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