Calculus I - Trig Functions

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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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.

$A unit circle with the angles $\frac{\pi }{6}$, $\frac{\pi }{4}$ and $\frac{\pi }{3}$ drawn in the first quadrant and labeled with the coordinates $\left( {\frac{{\sqrt 3 }}{2},\frac{1}{2}} \right)$, $\left( {\frac{{\sqrt 2 }}{2},\frac{{\sqrt 2 }}{2}} \right)$ and $\left( {\frac{1}{2},\frac{{\sqrt 3 }}{2}} \right)$ where they meet the circle. The axes are labeled with the angles 0 and $2\pi $, $\frac{\pi }{2}$, $\pi $ and $\frac{{3\pi }}{2}$ and the points (1,0), (0,1), (-1,0) and (0,-1). The angle $\frac{{3\pi }}{4}$ is drawn in the second quadrant and a dashed horizontal line connects the point where it meets the circle to the point for $\frac{\pi }{4}$, showing the two points have the same $y$-coordinate and $x$-coordinates that differ only in sign.$

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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,

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