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
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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 $270^\circ$. -
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 $60^\circ$. -
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.
