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What is the exact value of sin 240 degrees?

Understand the Problem

The question is asking for the exact value of sin 240 degrees, which is a trigonometric function value. To find this, we can identify the reference angle and the quadrant in which 240 degrees lies.

Answer

The exact value of $\sin(240^\circ)$ is $-\frac{\sqrt{3}}{2}$.
Answer for screen readers

The exact value of $\sin(240^\circ)$ is $-\frac{\sqrt{3}}{2}$.

Steps to Solve

  1. Identify the quadrant
    The angle of $240^\circ$ is located in the third quadrant of the unit circle because it is between $180^\circ$ and $270^\circ$.

  2. Determine the reference angle
    To find the reference angle, subtract $180^\circ$ (the angle at the start of the third quadrant) from $240^\circ$: $$ 240^\circ - 180^\circ = 60^\circ $$
    So, the reference angle is $60^\circ$.

  3. Identify the sine value in the third quadrant
    In the third quadrant, the sine function is negative. The sine of the reference angle $60^\circ$ is: $$ \sin(60^\circ) = \frac{\sqrt{3}}{2} $$
    Thus, in the third quadrant, $$ \sin(240^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2} $$

The exact value of $\sin(240^\circ)$ is $-\frac{\sqrt{3}}{2}$.

More Information

The sine function gives the y-coordinate of a point on the unit circle. The angle $240^\circ$ corresponds to a point in the third quadrant where both x and y are negative. Therefore, the sine value is negative.

Tips

  • A common mistake is forgetting that sine is negative in the third quadrant.
  • Another mistake is miscalculating the reference angle. Make sure to identify the starting point correctly when subtracting.
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