Introduction
What if the equation is ? Or ? These require identities, substitution, or clever factoring to solve.
Prerequisite Connection
You know how to solve basic trig equations and you know your trig identities.
Today's Increment
We're solving quadratic-form equations, using identities to simplify, and handling equations with multiple angles.
Why This Matters
These techniques are essential for Calculus, where optimization problems often lead to complex trig equations.
Advanced Techniques
Quadratic Form
If the equation looks like , substitute and solve the quadratic.
Identity Reduction
Use Pythagorean or Double-Angle identities to rewrite the equation in terms of a single trig function.
Multiple Angles
For , expand with the double-angle formula, factor, and solve. Remember: if , then .
Worked Examples
Example 1: Quadratic in Cosine
Solve on .
Substitute
Let . Equation becomes .
Factor
or
Back-Substitute
Answer:
Example 2: Using an Identity
Solve on .
Take Square Root
Solve for Both Cases
Answer:
Example 3: Multiple Angle (Advanced)
Solve on .
Expand Double Angle
Use .
Rearrange to Standard Form
(This is Example 1!)
Apply Previous Solution
We already solved this:
Answer:
Common Pitfalls
Forgetting ± when taking square roots
If , then . Both cases must be solved.
Extraneous solutions from squaring
If you square both sides of an equation, always check your answers in the original equation.
Real-World Application
Robotics & Inverse Kinematics
To position a robotic arm at a specific point in space, engineers must solve complex trig equations to determine the angles of each joint. These often involve quadratics in sine or cosine, requiring exactly the techniques learned in this lesson.
Practice Quiz
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