Cosine Function: Step-by-Step Guide
On the unit circle (a circle with radius 1), the cosine of an angle is the x-coordinate of the point where the angle's side meets the circle. We measure angles counterclockwise from the positive x-axis. In a right triangle, cosine is the adjacent side divided by the hypotenuse. Cosine repeats every 2 pi, and its values always stay between -1 and 1.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Bring the angle between 0 and 2 pi. Add or subtract full turns (2 pi) until the angle is between 0 and 2 pi.\[\theta' = \theta - 2\pi k,\;\; \theta' \in [0, 2\pi)\]
- Find the reference angle. The reference angle is the small, acute angle between the angle's side and the x-axis.\[\theta_{ref} = \begin{cases} \theta & Q1 \\ \pi - \theta & Q2 \\ \theta - \pi & Q3 \\ 2\pi - \theta & Q4 \end{cases}\]
- Find the sign from the quadrant. Cosine is positive in quadrants I and IV (where x is positive) and negative in quadrants II and III (where x is negative).\[\cos(\theta) > 0 \text{ in QI, QIV};\;\; \cos(\theta) < 0 \text{ in QII, QIII}\]
- Find the cosine of the reference angle. Use the unit circle values you have memorized for the common angles.\[\cos(0)=1,\;\cos(\pi/6)=\sqrt{3}/2,\;\cos(\pi/4)=\sqrt{2}/2,\;\cos(\pi/3)=1/2,\;\cos(\pi/2)=0\]
- Put the sign on. Take the value from the reference angle and give it the sign from the quadrant.\[\cos(\theta) = \pm \cos(\theta_{ref})\]
Worked example
Find cos(5*pi/4).
Step 1: 5 pi / 4 is already between 0 and 2 pi.
Step 2: 5 pi / 4 is in Quadrant III. The reference angle is 5 pi / 4 - pi = pi / 4.
Step 3: Cosine is negative in Quadrant III.
Step 4: cos(pi / 4) = root 2 / 2.
Step 5: Put the sign on: cos(5 pi / 4) = -root 2 / 2.
Check: On the unit circle, the point at 5 pi / 4 is (-root 2 / 2, -root 2 / 2). Its x-coordinate is -root 2 / 2.
Tips and common mistakes
- Cosine is an even function: cos(-theta) = cos(theta). A negative angle has the same cosine as the positive angle.
- Cosine is the x-coordinate on the unit circle. Think of how far left or right the point is.
- Cofunction identity: cos(theta) = sin(pi/2 - theta). This helps if you remember sine values better.
- cos(pi - theta) = -cos(theta).
- Cosine repeats every 2 pi: cos(theta + 2 pi k) = cos(theta) for any whole number k.
- Memorize cos(0), cos(pi/6), cos(pi/4), cos(pi/3), cos(pi/2) = 1, root 3 / 2, root 2 / 2, 1/2, 0.
- Cosine is always between -1 and 1. If you get something outside that, there is a mistake.