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

Trigonometry · Axiom Academy

EXAMPLE Solving a Trigonometric System Discover how unit circle constraints and algebraic techniques reveal the solutions Excellent work! You've successfully solved this complex trigonometric system. Here's what we learned: Unit Circle Recognition: Understanding that sin and cos represent coordinates on the unit circle is crucial for solving such systems Strategic Squaring: Squaring both equations and adding them allowed us to eliminate cross terms and use the Pythagorean identity Identity Recognition: Recognizing that sin(x)sin(y) + cos(x)cos(y) = cos(x - y) was key to progress Geometric Insight: The solution requires the two angles to differ by 90°, which has a beautiful geometric interpretation on the unit circle Verification Matters: Always check your solutions in the original equations to confirm they work! This problem demonstrates how algebraic manipulation, trigonometric identities, and geometric understanding work together to solve complex systems. The same techniques apply to many other trigonometric problems!

This is the written version of the interactive lesson above. See the full Trigonometry course.