Lesson 6.6

Equations of Quadratic Type

An expression like looks exactly like a quadratic in . Use substitution, factor, solve, and translate back to angles.

Introduction

Many trig equations are “quadratics in disguise.” If you replace with , the equation becomes a standard quadratic that you can factor or use the quadratic formula on — then substitute back and solve for the angles.

Past Knowledge

Quadratic factoring, the quadratic formula, and basic trig solving (6.2–6.5).

Today's Goal

Solve quadratic-type trig equations using u-substitution and the quadratic formula when needed.

Future Success

Combining identity substitution (6.7) with quadratic techniques handles the hardest problems in this chapter.

Key Concepts

The u-Substitution Strategy

Let (or cos, tan — whichever appears).
Rewrite as .
Factor or use the quadratic formula.
Substitute back: set .
Reject any root outside (range of sin/cos).
Find all angles on the required interval.

Range Check

After solving the quadratic, any value of that falls outside (for sine or cosine) gives no solution. For tangent, any real number is valid.

Worked Examples

Basic

Factorable Quadratic

Solve: on .

Step 1: Let :

Step 2: Factor:

Step 3: or . Both in ✓

Step 4:

Solution:

Intermediate

Using the Quadratic Formula

Solve: on .

Step 1: Not easily factorable. Use the quadratic formula with :

Step 2: ✓ (in )

✗ (outside range, reject)

Step 3: → use calculator: rad

Solution: radians

Advanced

Both Roots Rejected

Solve: .

Step 1: Quadratic formula:

Step 2: The discriminant is negative — the roots are complex. No real solutions.

Solution: No solution.

Common Pitfalls

Forgetting the Range Check

If the quadratic gives , that's outside and must be rejected. Students often try to find an angle for impossible values.

Real-Life Applications

Optics — Snell's Law Extended

When light passes through multiple layers of material, the refraction equations often result in quadratic-type trig equations. Optical engineers solve these to determine at which angles light will refract or undergo total internal reflection — discarding non-physical roots the same way you reject values outside .