Trigonometry Examples

$\sin (165)$

Step 1

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

$\sin (15)$

Step 2

Split

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

$\sin (45 - 30)$

Step 3

Separate negation.

$\sin (45 - (30))$

Step 4

Apply the difference of angles identity.

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

Step 5

The exact value of

$\sin (45)$ is

$\frac{\sqrt{2}}{2}$ .

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

Step 6

The exact value of

$\cos (30)$ is

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

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

Step 7

The exact value of

$\cos (45)$ is

$\frac{\sqrt{2}}{2}$ .

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

Step 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 9

Simplify

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply

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

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Step 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 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 9.1.1.3

Multiply

$2$ by

$3$ .

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

Step 9.1.1.4

Multiply

$2$ by

$2$ .

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

Step 9.1.2

Multiply

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

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Step 9.1.2.1

Multiply

$\frac{1}{2}$ by

$\frac{\sqrt{2}}{2}$ .

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

Step 9.1.2.2

Multiply

$2$ by

$2$ .

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

Step 9.2

Combine the numerators over the common denominator.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$