Lesson 6.5

Solving by Factoring

When an equation has a common trig factor, pull it out and apply the Zero Product Property — exactly as you did in algebra.

Introduction

If you see , resist the urge to divide by . Instead, factor out the common term and use the Zero Product Property to get allsolutions — including the ones you'd lose by dividing.

Past Knowledge

Zero Product Property from algebra, basic trig equation solving (6.2–6.4).

Today's Goal

Factor trig equations to find multiple solution pathways using the Zero Product Property.

Future Success

Quadratic-type factoring (6.6) extends this method to more complex trig expressions.

Key Concepts

The Factoring Strategy

Move all terms to one side (set equation = 0).
Factor out the GCF (greatest common factor) trig term.
Set each factor = 0 (Zero Product Property).
Solve each resulting equation independently.

⚠️ Never Divide by a Trig Function

Dividing both sides by (or any trig function) erases solutions where that function equals zero. Always factor instead. The solutions from the factor you would have divided away are often the ones students miss.

Worked Examples

Basic

GCF Factoring

Solve: on .

Step 1: Factor out :

Step 2: Set each factor = 0:

Solution:

Intermediate

Two-Term Factoring

Solve: on .

Step 1: Move to one side:

Step 2: Factor:

Step 3: Set each factor = 0:

Note: Check in the original equation — is undefined, so reject this solution.

Solution:

Advanced

What If You Divide?

Demonstrate: Show that dividing by in the basic example loses solutions.

Starting from , if we divide both sides by :

We get only 2 solutions — missing and (where ).

⚠️ Dividing lost half the solutions. Always factor instead.

Common Pitfalls

Dividing by a Trig Function

This erases zeros. Always factor instead of dividing.

Not Checking Domain Restrictions

When tangent appears, candidate solutions where tan is undefined must be rejected.

Real-Life Applications

Projectile Motion

In physics, the range equation for a projectile involves . Finding the launch angle for a specific range requires factoring trig equations — ignoring the zero-angle solution could mean missing the trivial case where the projectile goes straight up.