MathLabs

Problem 4

Find all real solutions of cos⁡2x+cos⁡22x+cos⁡23x=1\cos^2 x+\cos^2 2x+\cos^2 3x=1.
Step 5 of 5: Convert the cosine values into all real solutions
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x=π2+kπ,x=π4+kπ2,x=π6+kπ or 5π6+kπ,k∈Zx=\frac\pi2+k\pi,\quad x=\frac\pi4+\frac{k\pi}2,\quad x=\frac\pi6+k\pi\text{ or }\frac{5\pi}6+k\pi,\quad k\in\mathbb Z
Detailed analysis

Solving the three cosine conditions gives x=π/2+kπx=\pi/2+k\pi, x=π/4+kπ/2x=\pi/4+k\pi/2, and x=π/6+kπx=\pi/6+k\pi or x=5π/6+kπx=5\pi/6+k\pi, with k∈Zk\in\mathbb Z. These families exhaust the factored equation.