Multiple Angle Equations: Step-by-Step Guide
Some equations have a multiple of x inside the trig function, like sin(2x) = k or cos(3x + b) = k. To solve them, let u stand for the inside part, find every value of u in a longer range, then change back to find x.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Let u stand for the inside. Let u = nx (or nx + b), so the equation looks like sin(u) = k or cos(u) = k.\[u = n x + b\]
- Make the range for u longer. If x is between 0 and 2 pi, then u goes up to n times 2 pi (plus b). You need all the u answers in that longer range.\[x \in [0, 2\pi) \Rightarrow u \in [b, 2n\pi + b)\]
- Solve the basic equation for u. Find the first answers, then add 2 pi (or pi for tangent) over and over until you fill the longer range.\[u = u_0 + 2\pi k,\;\; u = (\pi - u_0) + 2\pi k\]
- Go back to x. For each u, find x = (u - b) / n.\[x = \frac{u - b}{n}\]
- Keep only the x values in range. Throw out any x that is outside the range the problem gives.\[x \in [0, 2\pi)\]
Worked example
Solve sin(2x) = sqrt(2)/2 on [0, 2*pi).
Step 1: Let u = 2x, so sin(u) = root 2 / 2.
Step 2: If x is between 0 and 2 pi, then u is between 0 and 4 pi.
Step 3: Basic answers for sin(u) = root 2 / 2: u = pi/4 and 3 pi / 4. Add 2 pi to each to get more: 9 pi / 4 and 11 pi / 4. So u = pi/4, 3 pi / 4, 9 pi / 4, 11 pi / 4.
Step 4: Divide by 2 to get x: x = pi/8, 3 pi / 8, 9 pi / 8, 11 pi / 8.
Step 5: All four are between 0 and 2 pi. The answers are x = pi/8, 3 pi / 8, 9 pi / 8, 11 pi / 8.
Check: sin(2 x pi/8) = sin(pi/4) = root 2 / 2. The others work too.
Tips and common mistakes
- When the inside is nx, you usually get about n times as many answers between 0 and 2 pi.
- Make the range longer BEFORE you look for all the u answers. Do not divide first.
- For cos(nx) = k, the u answers come in pairs: u0 and 2 pi - u0. Each pair keeps repeating.
- For tan(nx) = k, tangent repeats every pi, so add pi each time.
- If there is a shift, +b, in the inside, subtract it after you solve for u.
- Check that each x is in the range the problem asks for.
- Writing u down makes the algebra cleaner. Do not try to keep track of the multiplier in your head.
- A sketch of y = sin(2x) shows the faster wave and helps you count the answers.