Algebra Examples

sin (345)

$\sin (345)$

Step 1

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

345

$345$ can be split into

300 + 45

$300 + 45$ .

sin (300 + 45)

$\sin (300 + 45)$

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)

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

sin (300) cos (45) + cos (300) sin (45)

$\sin (300) \cos (45) + \cos (300) \sin (45)$

Step 3

Remove parentheses.

sin (300) cos (45) + cos (300) sin (45)

$\sin (300) \cos (45) + \cos (300) \sin (45)$

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

- sin (60) cos (45) + cos (300) sin (45)

$- \sin (60) \cos (45) + \cos (300) \sin (45)$

Step 4.2

The exact value of

sin (60)

$\sin (60)$ is

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

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

- \frac{\sqrt{3}}{2} cos (45) + cos (300) sin (45)

$- \frac{\sqrt{3}}{2} \cos (45) + \cos (300) \sin (45)$

Step 4.3

The exact value of

cos (45)

$\cos (45)$ is

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

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

- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + cos (300) sin (45)

$- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \cos (300) \sin (45)$

Step 4.4

Multiply

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

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

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Step 4.4.1

Multiply

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

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

$- \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \cos (300) \sin (45)$

Step 4.4.2

Combine using the product rule for radicals.

$- \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \cos (300) \sin (45)$

Step 4.4.3

Multiply

$2$ by

$3$ .

$- \frac{\sqrt{6}}{2 \cdot 2} + \cos (300) \sin (45)$

Step 4.4.4

Multiply

$2$ by

$2$ .

$- \frac{\sqrt{6}}{4} + \cos (300) \sin (45)$

Step 4.5

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

$\frac{1}{2}$ by

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

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

Step 4.8.2

Multiply

$2$ by

$2$ .

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

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Factor

$- 1$ out of

$- \sqrt{6}$ .

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

Step 5.3

Factor

$- 1$ out of

$\sqrt{2}$ .

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

Step 5.4

Factor

$- 1$ out of

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

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

Step 5.5

Simplify the expression.

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Step 5.5.1

Rewrite

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

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

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

Step 5.5.2

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$