Solving Equations Summary

Trigonometry · Axiom Academy

LESSON Solving Trigonometric Equations Let's review how to solve trigonometric equations using reference angles, identities, and careful attention to periodic solutions and domain restrictions. Isolate the Trig Function: Get one trigonometric function by itself on one side of the equation (e.g., sin( x ) = 1/2) Find Reference Solutions: Use the unit circle or inverse functions to find solutions in [0, 2π) or [0°, 360°) Check All Quadrants: Trigonometric functions typically have two solutions per period—identify which quadrants satisfy the equation Apply Periodicity: Add integer multiples of the period to generate all solutions (general solution) Multiple Angle Equations: For equations like sin(2 x ) = 1/2, substitute u = 2 x , solve for u , then divide by the coefficient Solution Count: An equation with coefficient n will have n times as many solutions in a given interval Identity Application: Use Pythagorean, double-angle, or other identities to transform complex equations into solvable forms Verify Solutions: Always check solutions in the original equation, especially when squaring or using identities that may introduce extraneous solutions Step-by-Step Example: Solving 2cos²( x ) − cos( x ) − 1 = 0 Recognize the form: This is a quadratic equation in cos( x ). Let u = cos( x ) to get 2 u ² − u − 1 = 0 Factor the quadratic: (2 u + 1)( u − 1) = 0, giving u = −1/2 or u = 1 Substitute back: cos( x ) = −1/2 or cos( x ) = 1 Solve cos( x ) = 1: This gives x = 0 in [0, 2π)

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