Cotangent Function: Step-by-Step Guide
Cotangent is the reciprocal of tangent: cot(theta) = cos(theta) / sin(theta) = 1 / tan(theta). It repeats every pi, and it is undefined wherever sin(theta) = 0 (at k pi).
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Reduce the angle by pi. Cotangent repeats every pi, so subtract multiples of pi to get an angle between 0 and pi.\[\cot(\theta) = \cot(\theta - k\pi)\]
- Find sine and cosine (or tangent). Find sin(theta) and cos(theta) using reference angles, or find tan(theta) first.\[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\]
- Check that sine is not zero. If sin(theta) = 0, cotangent is undefined.\[\sin(\theta) = 0 \Rightarrow \cot(\theta) \text{ undefined}\]
- Divide and simplify. Work out cos / sin, then simplify or rationalize.\[\cot(\theta) = \frac{1}{\tan(\theta)}\]
Worked example
Find cot(5*pi/6).
Step 1: 5 pi / 6 is already between 0 and pi.
Step 2: 5 pi / 6 is in Quadrant II. The reference angle is pi - 5 pi / 6 = pi / 6. cos(5 pi / 6) = -root 3 / 2 and sin(5 pi / 6) = 1/2.
Step 3: Sine is not zero, so cotangent is defined.
Step 4: cot(5 pi / 6) = (-root 3 / 2) / (1/2) = -root 3.
Check: tan(5 pi / 6) = -1 / root 3 = -root 3 / 3, and 1 / tan = -3 / root 3 = -root 3. It matches.
Tips and common mistakes
- cot(theta) = 1 / tan(theta) = cos(theta) / sin(theta).
- Cotangent is odd: cot(-theta) = -cot(theta).
- The period is pi, not 2 pi.
- There are asymptotes at k pi, where sine is zero.
- cot(pi/2) = 0, which is where tangent has its asymptote.
- Identity: 1 + cot^2(theta) = csc^2(theta).
- Cofunction identity: cot(theta) = tan(pi/2 - theta).
- Cotangent is positive in quadrants I and III and negative in II and IV, like tangent.