Question
Simplify \( \sin \left(240^{\circ}\right) \) without the use of a calculator. Write your answer in simplest surd form if necessary.
Ask by Cervantes Kelley. in South Africa
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \sin(240^\circ) = -\frac{\sqrt{3}}{2} \)
Solution
To simplify sin(240°), we first note that 240° can be expressed as 180° + 60°. Using the sine angle addition identity:
sin(180° + θ) = -sin(θ),
we substitute θ = 60°:
sin(240°) = sin(180° + 60°) = -sin(60°).
Since sin(60°) is √3/2, we have:
sin(240°) = -√3/2.
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Simplify this solution Beyond the Answer
To simplify \( \sin(240^{\circ}) \), we can first recognize that \( 240^{\circ} \) is in the third quadrant. The sine function is negative in the third quadrant. We can find the reference angle by calculating \( 240^{\circ} - 180^{\circ} = 60^{\circ} \). Using the sine value of the reference angle, we have: \[ \sin(240^{\circ}) = -\sin(60^{\circ}) \] We know from trigonometric values that \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \). Therefore, \[ \sin(240^{\circ}) = -\frac{\sqrt{3}}{2} \] Thus, the simplified form of \( \sin(240^{\circ}) \) is: \[ \boxed{-\frac{\sqrt{3}}{2}} \]
