Tangent Function: Step-by-Step Guide
Tangent is defined as tan(theta) = sin(theta) / cos(theta). In a right triangle, it is the opposite side divided by the adjacent side. Tangent repeats every pi (not 2 pi), and it is undefined wherever cos(theta) = 0, which happens at pi/2 + k pi.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Reduce the angle by whole multiples of pi. Since tangent repeats every pi, subtract multiples of pi to bring the angle between 0 and pi.\[\tan(\theta) = \tan(\theta - k\pi)\]
- Check for undefined values. If the angle is pi/2 + k pi, tangent is undefined (there is a vertical asymptote). Otherwise, continue.\[\tan\!\left(\tfrac{\pi}{2} + k\pi\right) \text{ undefined}\]
- Find the reference angle and sign. Find the reference angle like you did for sine and cosine. Tangent is positive in quadrants I and III and negative in II and IV.\[\tan(\theta) > 0 \text{ in QI, QIII};\;\; \tan(\theta) < 0 \text{ in QII, QIV}\]
- Find the tangent of the reference angle. Use tan = sin / cos or values you have memorized.\[\tan(0)=0,\;\tan(\pi/6)=\tfrac{\sqrt{3}}{3},\;\tan(\pi/4)=1,\;\tan(\pi/3)=\sqrt{3}\]
- Put the sign on. Combine the value from the reference angle with the sign from the quadrant.\[\tan(\theta) = \pm \tan(\theta_{ref})\]
Worked example
Find tan(2*pi/3).
Step 1: 2 pi / 3 is already between 0 and pi.
Step 2: 2 pi / 3 is not pi/2 + k pi, so tangent is defined.
Step 3: 2 pi / 3 is in Quadrant II. The reference angle is pi - 2 pi / 3 = pi / 3. Tangent is negative in Quadrant II.
Step 4: tan(pi / 3) = root 3.
Step 5: Put the sign on: tan(2 pi / 3) = -root 3.
Check: sin(2 pi / 3) = root 3 / 2 and cos(2 pi / 3) = -1/2, so tan = (root 3 / 2) / (-1/2) = -root 3.
Tips and common mistakes
- Tangent repeats every pi, not 2 pi. Reduce by pi to save time.
- Tangent is an odd function: tan(-theta) = -tan(theta).
- There are vertical asymptotes at pi/2 + k pi, where cosine is zero.
- tan(pi / 4) = 1. Memorize this one.
- Since tan = sin / cos, you can find tangent from the unit circle by dividing the y-coordinate by the x-coordinate.
- Tangent has no largest value. It gets bigger and bigger near the asymptotes.
- Cofunction identity: tan(theta) = cot(pi/2 - theta).
- In a triangle, tangent is opposite / adjacent. Be careful which side is which for your angle.