Exercise 2103Algebra › Complex numbersOlympiadAdded 3 Nov 2026

Solving a trigonometric equation via cubics

#trigonometric equation#cubic#cos

Statement

Solve the equation $4\cos^3\theta-3\cos\theta=1$ for $\theta\in[0,2\pi]$ .

Hint

Recognize $4\cos^3\theta-3\cos\theta=\cos3\theta$ .

Show solutionHide solution

Trigonometric identity. $\cos3\theta=4\cos^3\theta-3\cos\theta$ , so the equation is $\cos3\theta=1$ .

Solve. $3\theta\equiv0\pmod{2\pi}$ , so $\theta=2k\pi/3$ for $k=0,1,2$ .

Solutions in $[0,2\pi]$ . $\theta\in\{0,2\pi/3,4\pi/3\}$ .

Conclusion. $\theta\in\{0,2\pi/3,4\pi/3\}$ .