Trigonometry Examples

$\sin (165)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $165$ can be split into $120 + 45$ .

$\sin (120 + 45)$

Step 2

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

$\sin (120) \cos (45) + \cos (120) \sin (45)$

Step 3

Remove parentheses.

$\sin (120) \cos (45) + \cos (120) \sin (45)$

Step 4

Simplify each term.

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Step 4.1

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

$\sin (60) \cos (45) + \cos (120) \sin (45)$

Step 4.2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} \cos (45) + \cos (120) \sin (45)$

Step 4.3

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

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \cos (120) \sin (45)$

Step 4.4

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

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Step 4.4.1

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

$\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} + \cos (120) \sin (45)$

Step 4.4.2

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} + \cos (120) \sin (45)$

Step 4.4.3

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \cos (120) \sin (45)$

Step 4.4.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} + \cos (120) \sin (45)$

Step 4.5

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.

$\frac{\sqrt{6}}{4} - \cos (60) \sin (45)$

Step 4.6

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

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

Step 4.7

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

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

Step 4.8

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

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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:

$0.25881904 \dots$