Using Identities to Solve

Algebra 2 · Axiom Academy

LESSON Using Identities to Solve When an equation mixes sine and cosine, one identity rewrites it in a single function — and then it is just algebra. When an equation contains more than one trig function, there are too many moving parts to solve directly. Pick an identity that expresses one function through the other, substitute, and the equation collapses to a single function you can handle algebraically. Pythagorean — trade a cos² for a sin² Double-angle — split sin 2θ into sin θ · cos θ Solve on . It mixes a with a , so replace with and every term becomes sine — a plain quadratic you can factor. 3. A Double-Angle Substitution Solve on . The blocks a direct solve, so expand it with , move everything to one side, and factor out the common . Every problem like these follows the same four moves — the identity does the hard part, and the algebra is what you already know. Different trig functions, or a squared term? It can't be solved as written. Use a Pythagorean or double-angle identity to rewrite it in one function. Set it to zero and factor. Never divide by a variable factor; that deletes solutions. List all angles in from each factor — expect several. You turned two mixed trig equations into ordinary algebra by unifying each one with a single identity, then collecting every solution on . Scroll up to revisit any step.

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