Carey AP PrecalculusPractice Lab
Topic study & practice

3.10 · Trigonometric Equations and Inequalities

Understand the ideas before moving on. Explore distinctive knowledge notes, worked examples, and connections between representations.

All Unit 3 topicsPractice This Topic
GUIDED NOTES · TOPIC 3.10All 44 topicsExplore the interactive labs
UNDERSTAND · APPLY · EXPLAIN

Topic 3.10 · Trigonometric Equations and Inequalities

Find every valid trigonometric branch in the requested interval and preserve excluded asymptotes.

On 0≤x≤π/20\le x\le\pi/2, cos⁡x(2sin⁡x−1)=0\cos x(2\sin x-1)=0 has solutions x=π/6x=\pi/6 and x=π/2x=\pi/2. The marked zero labels the interval boundary; it is not a solution. Do not divide away a factor that can be zero. The number-line sketch is schematic; retain the stated interval and exact symbolic values.

Teaching illustration · scroll to inspect
A product equation splits into two factors; both resulting solutions are kept on the stated interval zero to pi over two.

What a strong response shows

  • Isolate the trig expression and solve for its angle before undoing an input transformation.

  • List all branches or intervals in one cycle, then shift them by the correct period.

  • Apply the ORIGINAL input interval and substitute endpoints; never include an undefined input.

Loading the interactive investigation…

01

What You Need to Know

Key idea

Trigonometric equations ask for inputs that produce a target value; inequalities ask for intervals where the function stays above or below a target.

Fast decision rule

Solve one cycle first, then include every solution in the requested interval.

Worked steps

Find the reference angle, select all matching quadrants, and use the interval to decide which periodic copies remain.

Common mistake

Stopping after the first solution.

Mini check

Why can one trig equation have more than one solution in one cycle?

02

Learning Objectives

Key idea

3.10.A — Solve equations and inequalities involving trigonometric functions.

Fast decision rule

For equations find boundary angles; for inequalities test one point in each interval between boundaries.

Worked steps

11. Solve the equality boundaries. 22. Mark them on a unit circle or number line. 33. Test signs. 44. Include or exclude endpoints correctly.

Common mistake

Including an endpoint for a strict inequality.

Mini check

What creates the intervals in a trig inequality?

03

Key Concepts

Key idea

Periodicity repeats solutions. Inequalities are sign-pattern problems between equality solutions.

Fast decision rule

Boundary first, test second, periodic copies last.

Worked steps

Mark all boundary angles in one cycle, choose a test angle in each region, and keep the regions with the desired sign.

Common mistake

Using equality answers alone for an inequality.

Mini check

What extra step does an inequality require?

04

Relevant Representations

Key idea

The unit circle organizes exact solutions; a graph or number line organizes solution intervals.

Equation-to-inequality workflow
StepEvidence
solve equalityboundary angles
mark intervalordered inputs
test regionssign of function
reportinterval notation or angles
Fast decision rule

Use the unit circle for angle location and the graph for interval behavior.

Worked steps

Translate a target trig value into unit-circle points, then project their angles onto the requested interval.

Common mistake

Reading graph intersections without checking the interval endpoints.

Mini check

Which representation makes periodic copies easiest to see?

05

Essential Relationships or Formulas

Key idea

The period determines how solutions repeat, while the unit circle determines one-cycle locations.

sin⁡(x+2π)=sin⁡x\sin(x+2\pi)=\sin x
cos⁡(x+2π)=cos⁡x\cos(x+2\pi)=\cos x
tan⁡(x+π)=tan⁡x\tan(x+\pi)=\tan x
Fast decision rule

Use the function’s period, not a memorized universal period.

Worked steps

List the one-cycle answers, add or subtract whole periods, and keep only values in the requested domain.

Common mistake

Adding a full turn to tangent when its repeat is shorter.

Mini check

Which period belongs to tangent?

06

Worked Examples

Key idea

A sign chart turns a trig inequality into a finite set of checks.

Fast decision rule

Find boundary inputs, split the interval, then test one interior point per region.

Worked steps

For a sine threshold, mark its two one-cycle crossings, test the arc between them, and repeat the interval pattern only as needed.

Common mistake

Testing a boundary point instead of an interior point.

Mini check

Why should a test point be inside an interval?

07

Common Mistakes

Key idea

Common errors are missing branches, wrong periods, and endpoint mistakes.

Fast decision rule

Check one cycle, period, interval, and endpoint symbols.

Worked steps

Substitute one reported solution back into the original equation or inequality.

Common mistake

Rounding angle boundaries so a true endpoint is lost.

Mini check

What should you do with exact angles when possible?

08

AP Exam Tips

Key idea

A unit-circle sketch is often faster and safer than a calculator table.

Practice

Use the Questionbank set below to test this skill.

Fast decision rule

Show one-cycle reasoning before listing periodic answers.

Worked steps

State the boundary angles, interval test, and final interval notation.

Common mistake

Listing decimals with no interval or endpoint evidence.

Mini check

Can another reader reproduce your solution set?

TRY IT IN A NEW SETTING

Can you transfer the method?

Solve 2cos⁡2x−cos⁡x=02\cos^2x-\cos x=0 on 0≤x<2π0\le x<2\pi, keeping every factor branch.

Compare your reasoning

Factor cos⁡x(2cos⁡x−1)=0\cos x(2\cos x-1)=0. Thus cos⁡x=0\cos x=0 or cos⁡x=1/2\cos x=1/2, giving x=π/2,3π/2,π/3,5π/3x=\pi/2,3\pi/2,\pi/3,5\pi/3. Dividing by cosine would lose two roots.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit