Solving 2cos²(x) - cos(x) - 1 = 0

Trigonometry · Axiom Academy

EXAMPLE Solving Quadratic Trigonometric Equations Learn to solve trigonometric equations using algebraic techniques and substitution. Excellent work! You've successfully solved a quadratic trigonometric equation. Here's what we learned: Recognition: When a trig equation contains terms like cos²(x) and cos(x), treat it as a quadratic in cos(x). Substitution Strategy: Let u = cos(x) (or u = sin(x), etc.) to transform the equation into standard algebraic form. Solve Algebraically: Factor or use the quadratic formula to find solutions for u, just like any other quadratic equation. Back-Substitution: Replace u with the original trig function and solve for x using unit circle values or inverse trig functions. Check the Interval: Always verify that your solutions fall within the specified domain (here [0, 2π)). Multiple Solutions: Trig equations often have multiple solutions due to periodicity - make sure to find all solutions in the given interval. This technique is powerful for solving various trigonometric equations that have polynomial structure. You can apply the same method to equations involving sin(x), tan(x), or other trig functions!

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