Analytic Trigonometry
Lessons in this chapter
About Analytic Trigonometry
Analytic Trigonometry covers every identity and equation-solving technique built from the core trig relationships.
Basic Trigonometric Identities introduces the fundamental relationships between the trig functions, before Simplifying and Proving Equivalence of Trigonometric Expressions uses them to rewrite and verify more complex expressions.
Sum and Difference Identities gives the formulas for the sine, cosine, and tangent of a sum or difference of two angles.
Trigonometric Equations and Inequalities closes the chapter by applying every identity to solve equations and inequalities involving trig functions.
The four lessons, from Basic Trigonometric Identities to Trigonometric Equations and Inequalities, each open as a practice worksheet in the page. Every problem is scored as it is answered, including the equation problems, where the usual loss is giving one solution and forgetting the others in the interval. Free to practice online.
Frequently asked questions
Basic Trigonometric Identities covers the Pythagorean, reciprocal, quotient, and even and odd identities. Simplifying and Proving Equivalence of Trigonometric Expressions uses them to rewrite one expression as another, Sum and Difference Identities adds the formulas for the sine, cosine, and tangent of a sum or difference of two angles, and Trigonometric Equations and Inequalities closes by solving for every angle in an interval rather than a single value. This sits in Functions (F) in the high school Common Core, and in Unit 3 of the AP Precalculus course framework, Trigonometric and Polar Functions.
The Pythagorean identity (sin²θ + cos²θ = 1) and the reciprocal and quotient identities relate the six trig functions to each other.
Simplify one side of an expression using known identities until it matches the other side exactly, or reduce both sides independently to the same simplified form.
The sum and difference identities give the sine, cosine, or tangent of a sum or difference of two angles in terms of the individual angles’ sine, cosine, and tangent values.
Use identities to rewrite the equation in terms of a single trig function, then solve like any other equation, checking the solution set against the interval given.
A trig equation's solutions are usually periodic, repeating every full cycle: the general solution needs a "+ 2πn" (or similar) term to capture every one, not just the solutions in one cycle.