Trigonometry Examples

\sin (15 °)

$\sin (15 °)$

Step 1

Split

15

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

\sin (45 - 30)

$\sin (45 - 30)$

Step 2

Separate negation.

\sin (45 - (30))

$\sin (45 - (30))$

Step 3

Apply the difference of angles identity.

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

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

Step 4

The exact value of

\sin (45)

$\sin (45)$ is

\frac{\sqrt{2}}{2}

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

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

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

Step 5

The exact value of

\cos (30)

$\cos (30)$ is

\frac{\sqrt{3}}{2}

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

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

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

Step 6

The exact value of

\cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

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

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

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

Step 7

The exact value of

\sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 8

Simplify

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

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

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply

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

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

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Step 8.1.1.1

Multiply

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ by

\frac{\sqrt{3}}{2}

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

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

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

Step 8.1.1.2

Combine using the product rule for radicals.

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

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

Step 8.1.1.3

Multiply

2

$2$ by

3

$3$ .

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

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

Step 8.1.1.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 8.1.2

Multiply

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

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

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Step 8.1.2.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{\sqrt{2}}{2}

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

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

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

Step 8.1.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 8.2

Combine the numerators over the common denominator.

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

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

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.25881904 \dots

$0.25881904 \dots$