What We Cover
Foundations
- Right-triangle trig (SOH-CAH-TOA)
- Degrees and radians
- Unit circle from scratch
- Reference angles
Trig Functions
- sin, cos, tan, sec, csc, cot
- Domain, range, period
- Graphing all six functions
- Amplitude, period, phase shift
Identities
- Pythagorean identities
- Sum and difference formulas
- Double angle, half angle
- Product-to-sum, sum-to-product
- Identity proof strategy
Equations
- Solving trig equations on intervals
- Inverse trig functions
- Quadratic-in-trig equations
Triangle Tools
- Law of sines
- Law of cosines
- Ambiguous case (SSA)
- Area formulas, Heron's formula
- Arc length and sector area
Complex & Polar
- Polar coordinates and graphs
- Complex numbers in polar form
- De Moivre's theorem
- Roots of complex numbers
How I'd Walk You Through a Trig Equation
Problem: Solve 2 sin²x − sin x − 1 = 0 on [0, 2π).
- See the quadratic shape. If u = sin x, the equation is 2u² − u − 1 = 0. The trig is a disguise — under it, this is plain algebra.
- Factor. (2u + 1)(u − 1) = 0 → u = −1/2 or u = 1.
- Translate back to trig. sin x = −1/2 or sin x = 1.
- Find every solution in the interval. sin x = 1 at x = π/2. sin x = −1/2 at x = 7π/6 and x = 11π/6. Total: 3 solutions.
- The pattern. Whenever you see a · f²(x) + b · f(x) + c = 0, substitute u = f(x), solve the quadratic, then translate back.
The real lesson: Trig equations are almost always algebra problems wearing a costume. Once you spot the algebra structure, the trig vocabulary becomes irrelevant to the hard step.
FAQ
Do I need to memorize the unit circle?
Eventually yes — but we teach you how to derive any unit circle value in 5 seconds from geometry, so the memorization becomes natural instead of forced.
I keep failing trig identity proofs. Help?
Yes — trig identity proofs are pattern recognition once you know the 3 core identities. We drill the patterns until they're second nature.
How does trig connect to calculus?
Trig functions are everywhere in calculus — limits, derivatives, integrals, related rates, polar coords. A strong trig foundation makes calc much easier.