Using Identities to Solve
When an equation mixes two different trig functions, use an identity to rewrite everything in terms of a single function — then solve with the techniques from Lessons 6.2–6.6.
Introduction
Equations like contain both sine and cosine. You can't isolate either one directly. The key move: substitute the Pythagorean identity to rewrite the entire equation in terms of sine alone — then factor or apply the quadratic formula.
Past Knowledge
All identity families (Chapter 5), quadratic-type solving (6.6), and factoring (6.5).
Today's Goal
Use Pythagorean and double-angle identities to reduce mixed-function equations to a single trig function.
Future Success
This technique is essential for solving trig equations in calculus and physics.
Key Concepts
Which Identity to Use?
| If You See… | Substitute… | To Get… |
|---|---|---|
| with sin terms | All sin | |
| with cos terms | All cos | |
| Product form (factor) | ||
| with sin terms | All sin |
The Decision Rule
Choose the identity that eliminates the minority function. If the equation has more sine terms, replace the cosine; if it has more cosine terms, replace the sine.
Worked Examples
Pythagorean Substitution
Solve: on .
Step 1: Replace with :
Step 2: Expand and rearrange:
Step 3: Factor:
Step 4: → ; →
Solution:
Double-Angle Substitution
Solve: on .
Step 1: Replace with :
Step 2: Move to one side and factor:
Step 3: → ; →
Solution:
cos 2θ with sin Terms
Solve: on .
Step 1: Use :
Step 2: Factor:
Step 3: → ; →
Solution:
Common Pitfalls
Choosing the Wrong cos 2θ Form
There are three forms of . If the other terms are in sine, use . If in cosine, use . Using the wrong form leaves you with two different trig functions — defeating the purpose.
Real-Life Applications
Mechanical Engineering — Harmonic Motion
Vibrating systems like car suspensions produce equations mixing and (or their squares). Engineers use identity substitution to reduce these to single-function equations, making it possible to find the resonant frequencies that could cause dangerous oscillations.
Practice Quiz
Loading...