Trigonometry Examples

\sin (285)

$\sin (285)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

285

$285$ can be split into

225 + 60

$225 + 60$ .

\sin (225 + 60)

$\sin (225 + 60)$

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)

$\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

\sin (225) \cos (60) + \cos (225) \sin (60)

$\sin (225) \cos (60) + \cos (225) \sin (60)$

Step 3

Remove parentheses.

\sin (225) \cos (60) + \cos (225) \sin (60)

$\sin (225) \cos (60) + \cos (225) \sin (60)$

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. Make the expression negative because sine is negative in the third quadrant.

- \sin (45) \cos (60) + \cos (225) \sin (60)

$- \sin (45) \cos (60) + \cos (225) \sin (60)$

Step 4.2

The exact value of

\sin (45)

$\sin (45)$ is

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

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

- \frac{\sqrt{2}}{2} \cos (60) + \cos (225) \sin (60)

$- \frac{\sqrt{2}}{2} \cos (60) + \cos (225) \sin (60)$

Step 4.3

The exact value of

\cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \cos (225) \sin (60)

$- \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \cos (225) \sin (60)$

Step 4.4

Multiply

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

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

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Step 4.4.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

- \frac{\sqrt{2}}{2 \cdot 2} + \cos (225) \sin (60)

$- \frac{\sqrt{2}}{2 \cdot 2} + \cos (225) \sin (60)$

Step 4.4.2

Multiply

2

$2$ by

2

$2$ .

- \frac{\sqrt{2}}{4} + \cos (225) \sin (60)

$- \frac{\sqrt{2}}{4} + \cos (225) \sin (60)$

- \frac{\sqrt{2}}{4} + \cos (225) \sin (60)

$- \frac{\sqrt{2}}{4} + \cos (225) \sin (60)$

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 third quadrant.

- \frac{\sqrt{2}}{4} - \cos (45) \sin (60)

$- \frac{\sqrt{2}}{4} - \cos (45) \sin (60)$

Step 4.6

The exact value of

\cos (45)

$\cos (45)$ is

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

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

- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \sin (60)

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \sin (60)$

Step 4.7

The exact value of

\sin (60)

$\sin (60)$ is

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

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

- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$

Step 4.8

Multiply

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

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

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Step 4.8.1

Multiply

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

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

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

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

- \frac{\sqrt{2}}{4} - \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 4.8.2

Combine using the product rule for radicals.

- \frac{\sqrt{2}}{4} - \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}

$- \frac{\sqrt{2}}{4} - \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}$

Step 4.8.3

Multiply

3

$3$ by

2

$2$ .

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

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

Step 4.8.4

Multiply

2

$2$ by

2

$2$ .

- \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}

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

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

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

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

Step 5.2

Factor

- 1

$- 1$ out of

- \sqrt{2}

$- \sqrt{2}$ .

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

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

Step 5.3

Factor

- 1

$- 1$ out of

- \sqrt{6}

$- \sqrt{6}$ .

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

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

Step 5.4

Factor

- 1

$- 1$ out of

- (\sqrt{2}) - (\sqrt{6})

$- (\sqrt{2}) - (\sqrt{6})$ .

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

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

Step 5.5

Simplify the expression.

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Step 5.5.1

Rewrite

- (\sqrt{2} + \sqrt{6})

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

- 1 (\sqrt{2} + \sqrt{6})

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

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

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

Step 5.5.2

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

- 0.96592582 \dots

$- 0.96592582 \dots$