Scroll to Top

Virtual Math Learning Center

Virtual Math Learning Center Texas A&M University Virtual Math Learning Center
tags:
Tangent Trigonometry Trigonometry Series Unit Circle

Trig Functions with the Unit Circle Exercise 1

Author: Hannah Solomon

The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Reveal Answer" to see the answer to the problem.

Problem:  Find the exact values of \(\quad\tan\left(\dfrac{3\pi}{4}\right),\quad\)  \(\tan(210^\circ),\quad \) and  \(\tan(\pi).\)

Answer: \(\tan\left(\dfrac{3\pi}{4}\right)=-1,\)  \(\tan(210^\circ) = -\dfrac{\sqrt{3}}{3},\) \(\tan(\pi) = 0\)

Solution Method: To find the exact value of tangent, we use the formula
    \begin{equation*}
        \tan(\theta)=\frac{\sin{\theta}}{\cos{\theta}} 
    \end{equation*}
So we can find the exact value of tangent by finding the exact values of sine and cosine. Using right triangles, we know that every \((x,y)\) point on the unit circle is equivalent to \((\cos{\theta},\sin{\theta})\). (see previous video) So to find the exact value of tangent at a special angle, we take the x and y coordinates from the unit circle and divide them. 

For \(\tan\left(\frac{3\pi}{4}\right),\) the coordinate at \(\frac{3\pi}{4}\) is \(\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right).\) So, 
    \begin{equation*} 
        \tan(\frac{3\pi}{4})= \frac{y}{x} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = \left(\frac{\sqrt{2}}{2}\right) \left(-\frac{2}{\sqrt{2}}\right) = -1
    \end{equation*}

For \(\tan(210^\circ),\) the coordinate is \(\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\). So,
    \begin{equation*}
        \tan(210^\circ)= \frac{y}{x} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \left( \frac{1}{2} \right) \left( \frac{2}{\sqrt{3}} \right) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
    \end{equation*}
We had to rationalize the fraction to get our final answer.

For \(\tan(\pi),\) \(\pi\) is a quadrantal angle located at (-1,0). So, 
    \begin{equation*}
        \tan(\pi) = \frac{y}{x} = \frac{0}{1} = 0
    \end{equation*}

Beware of Quadrantals: At the quadrantal angles \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), the x-coordinate is 0, meaning in the tangent ratio, we are dividing by 0. Dividing by 0 means the fraction is undefined! So \(\tan\left(\frac{\pi}{2}\right)\) and \(\tan\left(\frac{3\pi}{2}\right)\) are undefined.

See more videos from this section

Prev Video Next Video Return to Section