Trigonometry Examples

$\cos (45) \cos (150) - \sin (45) \sin (150)$

Step 1

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

$\frac{\sqrt{2}}{2} \cos (150) - \sin (45) \sin (150)$

Step 2

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.

$\frac{\sqrt{2}}{2} (- \cos (30)) - \sin (45) \sin (150)$

Step 3

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

$\frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2}) - \sin (45) \sin (150)$

Step 4

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

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Step 4.1

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

$- \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \sin (45) \sin (150)$

Step 4.2

Combine using the product rule for radicals.

$- \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \sin (45) \sin (150)$

Step 4.3

Multiply $2$ by $3$ .

$- \frac{\sqrt{6}}{2 \cdot 2} - \sin (45) \sin (150)$

Step 4.4

Multiply $2$ by $2$ .

$- \frac{\sqrt{6}}{4} - \sin (45) \sin (150)$

Step 5

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

$- \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \sin (150)$

Step 6

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

$- \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \sin (30)$

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $2$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$