Basic Trig Equations: Step-by-Step Guide
A basic trig equation looks like sin(x) = k, cos(x) = k, or tan(x) = k. To solve one, get the trig function by itself, use the inverse function to find one answer, then use the symmetry of the circle and repeating pattern to find all the answers in the range you want.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Get the trig function by itself. Move numbers around so one trig function equals one number.\[\sin(x) = k,\;\cos(x) = k,\;\text{or}\;\tan(x) = k\]
- For sin and cos, check that k is between -1 and 1. If k is bigger than 1 or less than -1 for sin or cos, there is no real solution. Tangent can equal any number.\[|k| \leq 1 \text{ for }\sin,\cos\]
- Find the first answer. Use arcsin, arccos, or arctan on your calculator.\[x_0 = \sin^{-1}(k),\;\cos^{-1}(k),\;\text{or}\;\tan^{-1}(k)\]
- Find the second answer between 0 and 2 pi. For sin(x) = k, the other answer is pi - x0. For cos(x) = k, it is 2 pi - x0. For tan(x) = k, add pi.\[\sin: x_0,\,\pi - x_0;\;\cos: x_0,\,2\pi - x_0;\;\tan: x_0,\,x_0 + \pi\]
- Add the repeating answers. Add 2 pi n (for sine and cosine) or pi n (for tangent) to include every answer. Then keep only the ones in the range you need.\[x = x_0 + 2\pi n,\;\; x = (\pi - x_0) + 2\pi n\]
Worked example
Solve 2*cos(x) - 1 = 0 on [0, 2*pi).
Step 1: 2 cos(x) - 1 = 0 becomes cos(x) = 1/2.
Step 2: 1/2 is between -1 and 1, so there are solutions.
Step 3: The first answer: x0 = arccos(1/2) = pi/3.
Step 4: The second answer: 2 pi - pi/3 = 5 pi / 3.
Step 5: Between 0 and 2 pi, the answers are x = pi/3 and x = 5 pi / 3.
Check: cos(pi/3) = 1/2 and cos(5 pi / 3) = 1/2. Both work.
Tips and common mistakes
- Always get the trig function by itself before using the inverse function.
- Sine and cosine equations usually have two answers between 0 and 2 pi. Tangent also has two, because it repeats every pi.
- Draw the unit circle to see where the matching answers are.
- If the problem says 'all solutions', add the + 2 pi n (or + pi n) part.
- Check each answer by putting it back in. It is easy to pick the wrong quadrant for the second answer.
- Be careful if you squared both sides. You might have a fake answer.
- If k is outside -1 to 1 for sin or cos, write 'no real solution'.
- Change between degrees and radians to match what the problem asks for.