Secant Function: Step-by-Step Guide
Secant is the reciprocal of cosine: sec(theta) = 1 / cos(theta). It is undefined wherever cos(theta) = 0 (at pi/2 + k pi). It repeats every 2 pi. The absolute value of secant is always at least 1.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Find cos(theta). First find the cosine of the angle using reference angles and the quadrant sign.\[\cos(\theta) = \pm \cos(\theta_{ref})\]
- Check that cosine is not zero. If cos(theta) = 0, secant is undefined.\[\cos(\theta) = 0 \Rightarrow \sec(\theta) \text{ undefined}\]
- Flip it. Take the reciprocal of cos(theta) to get sec(theta). Rationalize if needed.\[\sec(\theta) = \frac{1}{\cos(\theta)}\]
- Clean up. If there is a root in the bottom, multiply the top and bottom to get rid of it.\[\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}\]
Worked example
Find sec(5*pi/3).
Step 1: 5 pi / 3 is in Quadrant IV. The reference angle is 2 pi - 5 pi / 3 = pi / 3. Cosine is positive in Quadrant IV. cos(pi / 3) = 1/2, so cos(5 pi / 3) = 1/2.
Step 2: That is not zero, so secant is defined.
Step 3: sec(5 pi / 3) = 1 / (1/2) = 2.
Step 4: It is already simplified.
Check: sec^2 - tan^2 = 1. With tan(5 pi / 3) = -root 3, we get 4 - 3 = 1. It works.
Tips and common mistakes
- sec(theta) = 1 / cos(theta). Always find cosine first.
- Secant is even: sec(-theta) = sec(theta).
- The absolute value of secant is always 1 or more. It is never between -1 and 1.
- Secant repeats every 2 pi, like cosine.
- There are vertical asymptotes at pi/2, 3 pi / 2, and so on, where cosine is zero.
- Useful identity: 1 + tan^2(theta) = sec^2(theta).
- When rationalizing: sec(pi/6) = 1 / (root 3 / 2) = 2 / root 3 = 2 root 3 / 3.
- Do not mix up secant and cosecant. Secant goes with cosine. Cosecant goes with sine.