Sine Function: Step-by-Step Guide
On the unit circle (a circle with radius 1), the sine of an angle is the y-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, sine is the opposite side divided by the hypotenuse. Sine 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. If the angle is outside that range, add or subtract full turns (2 pi) to get an angle that points the same way.\[\theta' = \theta - 2\pi k,\;\; \theta' \in [0, 2\pi)\]
- Find the reference angle. The reference angle is the small, acute angle between the angle's side and the x-axis. How you find it depends on the quadrant.\[\theta_{ref} = \begin{cases} \theta & Q1 \\ \pi - \theta & Q2 \\ \theta - \pi & Q3 \\ 2\pi - \theta & Q4 \end{cases}\]
- Find the sign from the quadrant. Sine is positive in quadrants I and II (where y is positive) and negative in quadrants III and IV (where y is negative).\[\sin(\theta) > 0 \text{ in QI, QII};\;\; \sin(\theta) < 0 \text{ in QIII, QIV}\]
- Find the sine of the reference angle. Use the unit circle values for the common angles: 0, pi/6, pi/4, pi/3, and pi/2.\[\sin(0)=0,\;\sin(\pi/6)=1/2,\;\sin(\pi/4)=\sqrt{2}/2,\;\sin(\pi/3)=\sqrt{3}/2,\;\sin(\pi/2)=1\]
- Put the sign on. Take the value from the reference angle and give it the sign from the quadrant.\[\sin(\theta) = \pm \sin(\theta_{ref})\]
Worked example
Find sin(7*pi/6).
Step 1: 7 pi / 6 is already between 0 and 2 pi.
Step 2: 7 pi / 6 is in Quadrant III (between pi and 3 pi / 2). The reference angle is 7 pi / 6 - pi = pi / 6.
Step 3: Sine is negative in Quadrant III.
Step 4: sin(pi / 6) = 1/2.
Step 5: Put the sign on: sin(7 pi / 6) = -1/2.
Check: On the unit circle, the point at 7 pi / 6 is (-root 3 / 2, -1/2). Sine is the y-coordinate, which is -1/2.
Tips and common mistakes
- Memorize the values for 0, pi/6, pi/4, pi/3, and pi/2. All the other common sine values come from these.
- To remember the signs in each quadrant: 'All Students Take Calculus'. All are positive in I, Sine in II, Tangent in III, Cosine in IV.
- Sine is an odd function: sin(-theta) = -sin(theta). This makes negative angles quick.
- Sine repeats every 2 pi: sin(theta + 2 pi k) = sin(theta) for any whole number k.
- Sine is the y-coordinate. Picture how high the point is above the x-axis.
- sin(pi - theta) = sin(theta). Angles that add up to pi have the same sine.
- Sine is always between -1 and 1. If you get something outside that, there is a mistake.