Solving via Square Roots
When a trig equation involves a squared function like , isolate the square and take the square root — remembering the gives you solutions in multiple quadrants.
Introduction
Some trig equations involve a squared function but no linear term — think rather than . In these cases, isolate the squared expression and take the square root directly, generating two equations to solve.
Past Knowledge
Basic solving (6.2), general solutions (6.3), and algebraic square-root property.
Today's Goal
Solve equations of the form using the square-root property.
Future Success
Recognizing when to use square roots vs. factoring (6.5) is a key decision point in harder problems.
Key Concepts
The Square Root Method
- Isolate the squared trig function:
- Take the square root of both sides:
- Solve two separate equations: and
Why ± Matters Here
The is essential because squaring erases sign information. For example, means or — giving solutions in all four quadrants.
When to Use This Method
Use square roots when the equation has a squared trig term but no linear trig term. If both and appear, use factoring (Lesson 6.5–6.6) instead.
Worked Examples
Simple Square Root
Solve: on .
Step 1: Isolate:
Step 2: Take square root:
Step 3: Solve each:
- →
- →
Solution:
Tangent Squared
Solve: on .
Step 1: Take square root:
Step 2: →
→
Solution:
No Solution Case
Solve: .
Step 1: Isolate:
Step 2: A square is never negative. There is no solution.
Solution: No solution (the equation is impossible).
Common Pitfalls
Dropping the Negative Root
When you take , you get . Forgetting the negative root cuts your solutions in half.
Real-Life Applications
Structural Engineering — Load Analysis
When analyzing forces on a bridge, engineers often encounter squared trig expressions from vector decomposition. Solving determines the angles at which structural members carry specific load components — and both positive and negative roots correspond to physically meaningful directions.
Practice Quiz
Loading...