Question

Find the exact value of sin240°.

Answer 1

Answer:

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

Step-by-step explanation:

We need to find the value of 240°

To find the value, first we need to understand in which quadrant 240° lies.

We know that there are four quadrants.

First quadrant lies from $(0,\:\frac{\pi }{2})$ in radians or 0 to 90° in degrees.

Second quadrant lies from $(\frac{\pi }{2},\:\pi)$ in radians or 90° to 180° in degrees.

Third quadrant lies from $(\pi,\:\frac{3\pi }{2})$ in radians or  180° to 270° in degrees.

Fourth quadrant lies from $(\frac{3\pi }{2},\:}2\pi )$ in radians or  270° to 360° in degrees.

Clearly we can see that 240° lies in the third quadrant, and in the third quadrant sin theta is negative.

Now, $\text{sin\:240}^{\circ}=\text{sin\:(180+60)}^{\circ}$

We know that $\text{sin}(\pi +x)=-\text{sin}x$

So,  $\text{sin\:240}^{\circ}=-\text{sin\:60}^{\circ}$

Hence, $\text{sin\:240}^{\circ}=-\frac{\sqrt{3}}{2}$

Answer 2

$\sin240^\circ=\sin(180-240)^\circ=\sin(-60^\circ)=-\sin60^\circ=-\dfrac{\sqrt3}2$

using the fact that (1) $\sin x=\sin(\pi -x)$ (where $x$ is in radians; replace with $180^\circ$ to convert to degrees) and (2) $\sin(-x)=-\sin x$ for all $x$. Then $\sin\dfrac\pi3=\sin60^\circ$ is a known value.