Section 1.3 : Trig Functions
2. Determine the exact value of \(\displaystyle \sin \left( { - \frac{{4\pi }}{3}} \right)\) without using a calculator.
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First we can notice that \( - \pi - \frac{\pi }{3} = - \frac{{4\pi }}{3}\) and so (remembering that negative angles are rotated clockwise) we can see that the terminal line for \( - \frac{{4\pi }}{3}\) will form an angle of \(\frac{\pi }{3}\) with the negative \(x\)-axis in the second quadrant and we’ll have the following unit circle for this problem.
The coordinates of the line representing \( - \frac{{4\pi }}{3}\) will be the same as the coordinates of the line representing \(\frac{\pi }{3}\) except that the \(x\) coordinate will now be negative. So, our new coordinates will then be \(\left( { - \frac{1}{2},\frac{{\sqrt 3 }}{2}} \right)\) and so the answer is,
\[\sin \left( { - \frac{{4\pi }}{3}} \right) = \frac{{\sqrt 3 }}{2}\]