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Section 1.3 : Trig Functions

8. Determine the exact value of \(\displaystyle \tan \left( { - \frac{\pi }{3}} \right)\) without using a calculator.

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Hint : Even though a unit circle only tells us information about sine and cosine it is still useful for tangents so sketch a unit circle and relate the angle to one of the standard angles in the first quadrant.
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To do this problem all we need to notice is that \( - \frac{\pi }{3}\) will form an angle of \(\frac{\pi }{3}\) with the positive \(x\)-axis in the fourth quadrant and we’ll have the following unit circle for this problem.

Hint : Given the obvious symmetry in the unit circle relate the coordinates of the line representing \( - \frac{\pi }{3}\) to the coordinates of the line representing \(\frac{\pi }{3}\) and use the definition of tangent in terms of sine and cosine to answer the question.
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The coordinates of the line representing \( - \frac{\pi }{3}\) will be the same as the coordinates of the line representing \(\frac{\pi }{3}\) except that the \(y\) 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,

\[\tan \left( -\frac{\pi }{3} \right)=\frac{\sin \left( -\frac{\pi }{3} \right)}{\cos \left( -\frac{\pi }{3} \right)}=\frac{-{}^{\sqrt{3}}/{}_{2}}{{}^{1}/{}_{2}}=-\sqrt{3}\]