Cosecant Function: Step-by-Step Guide
Cosecant is the reciprocal of sine: csc(theta) = 1 / sin(theta). It is undefined wherever sin(theta) = 0 (at k pi). It repeats every 2 pi. The absolute value of cosecant is always at least 1.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Find sin(theta). Find the sine of the angle using reference angles and the quadrant sign.\[\sin(\theta) = \pm \sin(\theta_{ref})\]
- Check that sine is not zero. If sin(theta) = 0, cosecant is undefined.\[\sin(\theta) = 0 \Rightarrow \csc(\theta) \text{ undefined}\]
- Flip it. Take the reciprocal of sin(theta) to get csc(theta).\[\csc(\theta) = \frac{1}{\sin(\theta)}\]
- Clean up. If there is a root in the bottom, rationalize it.\[\frac{1}{\sqrt{a}/b} = \frac{b\sqrt{a}}{a}\]
Worked example
Find csc(7*pi/4).
Step 1: 7 pi / 4 is in Quadrant IV. The reference angle is 2 pi - 7 pi / 4 = pi / 4. Sine is negative in Quadrant IV. sin(pi / 4) = root 2 / 2, so sin(7 pi / 4) = -root 2 / 2.
Step 2: That is not zero, so cosecant is defined.
Step 3: csc(7 pi / 4) = 1 / (-root 2 / 2) = -2 / root 2.
Step 4: Rationalize: -2 / root 2 x (root 2 / root 2) = -2 root 2 / 2 = -root 2.
Check: csc^2 - cot^2 = 1. With cot(7 pi / 4) = -1, we get 2 - 1 = 1. It works.
Tips and common mistakes
- csc(theta) = 1 / sin(theta). Always find the sine first.
- Cosecant is odd: csc(-theta) = -csc(theta).
- The absolute value of cosecant is always 1 or more. It is never between -1 and 1.
- Cosecant repeats every 2 pi, just like sine.
- There are vertical asymptotes at 0, pi, 2 pi, and so on, where sine is zero.
- Useful identity: 1 + cot^2(theta) = csc^2(theta).
- Cofunction identity: csc(theta) = sec(pi/2 - theta).
- Common values: csc(pi/6) = 2, csc(pi/4) = root 2, csc(pi/3) = 2 root 3 / 3, csc(pi/2) = 1.