Expand Using Sum/Difference Formulas cos(135+30)

$\cos (135 + 30)$

Step 1

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

$\cos (30) \cdot \cos (135) - \sin (30) \cdot \sin (135)$

Step 2

Remove parentheses.

$\cos (30) \cdot \cos (135) - \sin (30) \cdot \sin (135)$

Step 3

Simplify each term.

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

$\frac{\sqrt{3}}{2} \cdot \cos (135) - \sin (30) \cdot \sin (135)$

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.

$\frac{\sqrt{3}}{2} \cdot (- \cos (45)) - \sin (30) \cdot \sin (135)$

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

$\frac{\sqrt{3}}{2} \cdot (- \frac{\sqrt{2}}{2}) - \sin (30) \cdot \sin (135)$

Multiply $\frac{\sqrt{3}}{2} (- \frac{\sqrt{2}}{2})$ .

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

$- \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \sin (30) \cdot \sin (135)$

Combine using the product rule for radicals.

$- \frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \sin (30) \cdot \sin (135)$

Multiply $3$ by $2$ .

$- \frac{\sqrt{6}}{2 \cdot 2} - \sin (30) \cdot \sin (135)$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6}}{4} - \sin (30) \cdot \sin (135)$

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

$- \frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \sin (135)$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \sin (45)$

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

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

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

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

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

Multiply $2$ by $2$ .

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

Step 4

Simplify.

Combine the numerators over the common denominator.

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

Factor $- 1$ out of $- \sqrt{6}$ .

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

Factor $- 1$ out of $- \sqrt{2}$ .

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

Factor $- 1$ out of $- (\sqrt{6}) - (\sqrt{2})$ .

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$

Expand Using Sum/Difference Formulas cos(135+30)

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