Finding All Solutions to cos(x) = -√2/2

Trigonometry · Axiom Academy

EXAMPLE Finding All Solutions to cos(x) = -√2/2 Learn to find specific solutions in [0, 2π) and write the general solution formula by identifying reference angles and quadrants. Excellent work! You've successfully found all solutions to the trigonometric equation. Here's what we learned: Sign Analysis: First identify where the trig function has the correct sign. Cosine is negative in Quadrants II and III. Reference Angle: Find the reference angle using the positive value: cos(θ) = √2/2 gives θ = π/4. Specific Solutions: Use the reference angle to find solutions in each appropriate quadrant. For Quadrant II: π - π/4 = 3π/4. For Quadrant III: π + π/4 = 5π/4. General Solution: Add 2πn to each specific solution to account for the periodic nature of cosine: x = 3π/4 + 2πn and x = 5π/4 + 2πn, where n is any integer. Verification: Always check that your solutions satisfy the original equation and fall within the required interval. This systematic approach works for any trigonometric equation. Remember: analyze signs, find the reference angle, locate solutions in the correct quadrants, then generalize with the period!

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