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### Section 1.3 : Trig Functions

5. Determine the exact value of $$\displaystyle \tan \left( {\frac{{3\pi }}{4}} \right)$$ 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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First we can notice that $$\pi - \frac{\pi }{4} = \frac{{3\pi }}{4}$$ and so (remembering that negative angles are rotated clockwise) we can see that the terminal line for $$\frac{{3\pi }}{4}$$ will form an angle of $$\frac{\pi }{4}$$ with the negative $$x$$-axis in the second 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 $$\frac{{3\pi }}{4}$$ to the coordinates of the line representing $$\frac{\pi }{4}$$ and and then recall how tangent is defined in terms of sine and cosine to answer the question.
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The coordinates of the line representing $$\frac{{3\pi }}{4}$$ will be the same as the coordinates of the line representing $$\frac{\pi }{4}$$ except that the $$x$$ coordinate will now be negative. So, our new coordinates will then be $$\left( { - \frac{{\sqrt 2 }}{2},\frac{{\sqrt 2 }}{2}} \right)$$ and so the answer is,

$\tan \left( \frac{3\pi }{4} \right)=\frac{\sin \left( \frac{3\pi }{4} \right)}{\cos \left( \frac{3\pi }{4} \right)}=\frac{{}^{\sqrt{2}}/{}_{2}}{-{}^{\sqrt{2}}/{}_{2}}=-1$