Expand Using Sum/Difference Formulas sin(240)

$\sin (240)$

The angle $240$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\sin (240 + 0)$

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

$\sin (240) \cos (0) + \cos (240) \sin (0)$

Remove parentheses.

$\sin (240) \cos (0) + \cos (240) \sin (0)$

Simplify each term.

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.

$- \sin (60) \cos (0) + \cos (240) \sin (0)$

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

$- \frac{\sqrt{3}}{2} \cos (0) + \cos (240) \sin (0)$

The exact value of $\cos (0)$ is $1$ .

$- \frac{\sqrt{3}}{2} \cdot 1 + \cos (240) \sin (0)$

Multiply $- 1$ by $1$ .

$- \frac{\sqrt{3}}{2} + \cos (240) \sin (0)$

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.

$- \frac{\sqrt{3}}{2} - \cos (60) \sin (0)$

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

$- \frac{\sqrt{3}}{2} - \frac{1}{2} \sin (0)$

The exact value of $\sin (0)$ is $0$ .

$- \frac{\sqrt{3}}{2} - \frac{1}{2} \cdot 0$

Multiply $- \frac{1}{2} \cdot 0$ .

Multiply $0$ by $- 1$ .

$- \frac{\sqrt{3}}{2} + 0 (\frac{1}{2})$

Multiply $0$ by $\frac{1}{2}$ .

$- \frac{\sqrt{3}}{2} + 0$

Add $- \frac{\sqrt{3}}{2}$ and $0$ .

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.86602540 \dots$

Expand Using Sum/Difference Formulas sin(240)

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