Quadratic Trig Equations: Step-by-Step Guide
Some trig equations look like a quadratic: a x f(x)^2 + b x f(x) + c = 0, where f(x) is one trig function. To solve them, let u stand for the trig function, solve the quadratic for u, then go back and solve each simple trig equation.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Use only one trig function. Use the Pythagorean identities (like sin^2 + cos^2 = 1) so the equation has just one kind of trig function.\[\sin^2 x = 1 - \cos^2 x\]
- Let u be the trig function. Replace the trig function with u. Now it is a regular quadratic equation.\[a u^2 + b u + c = 0\]
- Solve the quadratic. Factor it or use the quadratic formula to find u.\[u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- Throw out impossible values. If u stands for sin or cos, it must be between -1 and 1. Drop any value that is not.\[|u| \leq 1 \text{ for sin, cos}\]
- Solve for x. For each good u, solve the trig equation (using inverse functions and the repeating pattern) to find every x in the range.\[x = f^{-1}(u) + 2\pi n\]
Worked example
Solve 2*sin^2(x) - sin(x) - 1 = 0 on [0, 2*pi).
Step 1: The equation already has only one trig function, sine.
Step 2: Let u = sin(x). Then 2u^2 - u - 1 = 0.
Step 3: Factor: (2u + 1)(u - 1) = 0, so u = -1/2 or u = 1.
Step 4: Both values are between -1 and 1, so we keep both.
Step 5: sin(x) = 1 gives x = pi/2. sin(x) = -1/2 gives x = 7 pi / 6 or x = 11 pi / 6.
The answers are x = pi/2, 7 pi / 6, and 11 pi / 6.
Check: sin(pi/2) = 1, so 2(1) - 1 - 1 = 0. sin(7 pi / 6) = -1/2, so 2(1/4) - (-1/2) - 1 = 1/2 + 1/2 - 1 = 0.
Tips and common mistakes
- If you see a squared trig function, try using u for the trig function.
- Use the Pythagorean identities to change a mixed equation into one with a single trig function.
- Before you go back to x, check that u is between -1 and 1 for sin and cos.
- Factor if you can. Use the quadratic formula if you cannot.
- Each good u usually gives two x values in one trip around the circle.
- Put all the answers together and put them in order.
- A double root gives just one answer, not two.
- Check each answer in the original equation.