Solving sin(x) + cos(x) = 1

Trigonometry · Axiom Academy

EXAMPLE Solving sin(x) + cos(x) = 1 Learn the squaring method and discover why checking for extraneous solutions is crucial! Step 11: Verification - Identifying Valid Solutions We found four potential solutions. Now we must check each one in the original equation sin(x) + cos(x) = 1. Click on each solution to verify it: Excellent work! You've successfully solved a trigonometric equation using the squaring method and verified all solutions. Here's what we learned: Squaring Can Introduce Extraneous Solutions: When we square both sides of an equation, we may introduce solutions that don't satisfy the original equation. Always verify! Pythagorean Identity is Essential: The identity sin²(x) + cos²(x) = 1 allowed us to convert everything to cosine terms. Factoring Reveals Multiple Cases: After simplifying, we got 2cos(x)(cos(x) - 1) = 0, giving us cos(x) = 0 or cos(x) = 1. Verification is Non-Negotiable: In this problem, x = 0 and x = π/2 are valid solutions, while x = 3π/2 and x = 2π are extraneous. Check in the Original Equation: Always substitute back into the original equation, not the squared version! The squaring method is powerful but requires careful verification. This technique applies to many types of equations—whenever you square, cube, or apply other operations that aren't one-to-one, check your solutions!

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