Trigonometric equations and inequalities — AP Precalculus
1. Overview of Trigonometric Equations and Inequalities ★★☆☆☆ ⏱ 3 min
Trigonometric equations and inequalities are statements involving trigonometric functions that require finding all input values that satisfy the given relationship. Unlike polynomial or rational equations, trigonometric functions are periodic, meaning most non-restricted problems have infinitely many solutions.
On the AP Precalculus exam, questions almost always specify a restricted domain (most commonly $0 \leq x < 2\pi$), but may also ask for a general solution or to count the number of solutions in a given interval. This topic accounts for ~12% of Unit 3 and appears in both multiple-choice and free-response sections.
2. Solving Linear Trigonometric Equations ★★★☆☆ ⏱ 4 min
The core solving strategy follows four key steps: 1. Isolate the trigonometric term on one side of the equation; 2. Check if the resulting right-hand side is within the range of the trigonometric function; 3. Find reference solutions for the argument using the unit circle or inverse trigonometry; 4. Use symmetry to find all solutions for the argument, solve for $x$, then filter to the required domain.
Exam tip: Always check the range of the isolated trigonometric term first—if it's outside the possible range (e.g., $\cos x = 2$), you can immediately conclude no real solutions exist, saving time on multiple-choice questions.
3. Solving Quadratic Trigonometric Equations ★★★☆☆ ⏱ 4 min
To solve these equations, first use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to convert mixed terms into a single-variable quadratic. Next, solve for the trigonometric function by factoring or the quadratic formula. Each solution becomes a separate linear trigonometric equation, which you solve using the linear method.
Exam tip: When you use an operation that can introduce extraneous solutions (like squaring both sides of an equation), always verify your solutions by plugging them back into the original equation.
4. Solving Trigonometric Inequalities ★★★★☆ ⏱ 3 min
A trigonometric inequality requires finding all input values that make a trigonometric expression satisfy an inequality relationship. We leverage continuity and periodicity of trigonometric functions to find valid solution intervals.
The core workflow is: 1. Solve the corresponding equality (replace the inequality with an equals sign) to find all critical points; 2. Sort critical points in increasing order; 3. Test the sign of the expression in each interval between critical points, or use unit circle symmetry to identify the valid region directly.
Exam tip: Remember that cosine corresponds to the $x$-coordinate and sine to the $y$-coordinate on the unit circle, so you can immediately identify which quadrants satisfy the inequality without testing every interval.
Common Pitfalls
Why: Students often only add the periodicity for the original trigonometric function and forget to adjust for the horizontal stretch.
Why: Students stop after finding the reference angle from the inverse trigonometric function, and forget sine and cosine take each value twice per period.
Why: Students memorize $2\pi$ periodicity from sine and cosine, and forget tangent has a shorter period of $\pi$.
Why: Students focus on factoring the quadratic and forget to check if each solution is within the range of the trigonometric function.
Why: Students rush to write the solution and don't check the original inequality sign.
Why: Squaring introduces solutions that make the two sides negatives of each other, which satisfy the squared equation but not the original.