Solving with Multiple Angles
When the equation is instead of , the argument is compressed. You need to solve in a wider interval first, then divide back to get your final answers.
Introduction
When the argument of a trig function is , , or , the function completes its cycle faster (or slower). To find all solutions for on , you first solve for the inner argument on a scaled interval, then divide back.
Past Knowledge
General solutions (6.3), period of trig functions, and all solving techniques (6.2–6.7).
Today's Goal
Solve equations with arguments like or by widening the interval, then dividing back.
Future Success
This technique combines all Chapter 6 skills into the most complex type of trig equation solving.
Key Concepts
The Interval Scaling Rule
If and the argument is , then solve for on first.
| Argument | θ Interval | Solve on | Expected # Solutions |
|---|---|---|---|
| Up to 4 | |||
| Up to 6 | |||
| Up to 1 |
The 4-Step Process
- Let and determine the scaled interval for .
- Solve the basic equation on the scaled interval.
- Divide each solution by to get .
- Verify each is in the original interval .
Why More Solutions?
If the argument is , the function completes two full cycles as goes from to . Each cycle contributes its own set of solutions, so you typically get twice as many answers.
Worked Examples
Solving sin 2θ = ½
Solve: on .
Step 1: Let . Since , .
Step 2: Solve on :
Step 3: Divide by 2:
Solution:
Solving cos 3θ = 0
Solve: on .
Step 1: Let , so .
Step 2: → . On :
Step 3: Divide by 3:
Solution: 6 solutions —
Half-Angle Argument
Solve: on .
Step 1: Let , so .
Step 2: on →
Step 3: Multiply by 2:
Solution: (only 1 solution!)
Common Pitfalls
Not Widening the Interval
If you solve on instead of , you lose half the solutions.
Dividing the Argument Before Solving
does not mean . You cannot divide inside the function — treat as a single input.
Real-Life Applications
Electrical Engineering — AC Circuit Harmonics
Power grids carry not just the fundamental 60 Hz signal, but also harmonics at 120 Hz (), 180 Hz (), etc. Engineers solve equations with multiple-angle arguments to identify at which points in the cycle these harmonics hit critical voltage levels, using exactly the “widen the interval, then divide back” technique.
Practice Quiz
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