Lesson 6.2

Solving Basic Trig Equations

Isolate the trig function, read the unit circle, and write the angle. Solving trig equations begins with the same “undo” logic you used in algebra — just with sine, cosine, or tangent instead of .

Introduction

In algebra, is solved by isolating . Here, we do the same — but we isolate , , or , then use the unit circle to find the angles.

Past Knowledge

Restricted domains (6.1), unit circle values, and algebraic equation-solving.

Today's Goal

Solve equations by isolating a single trig function and finding all solutions on a given interval.

Future Success

More advanced solving strategies (factoring, identities) all build on this basic isolation technique.

Key Concepts

The 3-Step Process

  1. Isolate the trig function on one side (treat as a single variable).
  2. Identify the reference angle from the unit circle.
  3. Find all angles that work in the given interval, using quadrant analysis.

Which Quadrants?

If…Solutions in Quadrants
QI, QII
QIII, QIV
QI, QIV
QII, QIII
QI, QIII
QII, QIV

Don't Forget the Second Solution!

Most basic trig equations on have two solutions — one from each valid quadrant. Always check both.

Worked Examples

Basic

Solving for Sine

Solve: on .

Step 1: Isolate:

Step 2: Reference angle: , and sine is positive → QI, QII.

Step 3: (QI) and (QII).

Solution:

Intermediate

Solving for Cosine (Negative Value)

Solve: on .

Step 1: Isolate:

Step 2: Reference angle: , and cosine is negative → QII, QIII.

Step 3: (QII) and (QIII).

Solution:

Advanced

Multi-Step Isolation

Solve: on .

Step 1: Isolate:

Step 2: Reference angle: , and tangent is negative → QII, QIV.

Step 3: (QII) and (QIV).

Solution:

Common Pitfalls

Forgetting the Second Quadrant Solution

On , most basic equations have two solutions. If you only find one, you've likely missed the solution in the other valid quadrant.

Real-Life Applications

Daylight Hours Prediction

The number of daylight hours over a year follows a sinusoidal pattern. Climatologists solve equations like to predict exactly which day of the year a city will have 14 hours of daylight — the same isolation-and-solve technique you're learning here.

Practice Quiz

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