Solving Trigonometric Equations

Solving trigonometric equations involves finding the unknown angle(s) that make the equation true. In advanced trigonometry, these equations often look like algebraic equations (such as quadratics) or contain multiple different angles, requiring you to use trigonometric identities to simplify them first.

Quadratic-Form Trigonometric Equations

Sometimes, a trigonometric equation takes the form of a quadratic equation. You can solve these by factoring, just as you would with a regular polynomial.

Example: Solve $2\cos^2 x - \cos x - 1 = 0$ on the interval $[0, 2\pi)$.

1. Treat the trigonometric function as a variable. Imagine $u = \cos x$. The equation becomes $2u^2 - u - 1 = 0$.
2. Factor the quadratic. $(2u + 1)(u - 1) = 0$.
3. Substitute the trig function back in. $(2\cos x + 1)(\cos x - 1) = 0$.
4. Set each factor to zero.
   • $2\cos x + 1 = 0 \implies \cos x = -\frac{1}{2}$
   • $\cos x - 1 = 0 \implies \cos x = 1$
5. Find the angles on $[0, 2\pi)$.
   • For $\cos x = -\frac{1}{2}$, the solutions are $x = \frac{2\pi}{3}$ and $x = \frac{4\pi}{3}$.
   • For $\cos x = 1$, the solution is $x = 0$.

The final solutions are $x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}$.

Equations with Multiple Angles

If an equation contains different arguments (like $2x$ and $x$), you must use trigonometric identities to rewrite the equation so that all trig functions have the same angle.

Example: Solve $\sin 2x = \cos x$ on the interval $[0, 2\pi)$.

1. Use the double-angle identity. We know that $\sin 2x = 2\sin x \cos x$.
2. Rewrite the equation. $2\sin x \cos x = \cos x$.
3. Move all terms to one side and factor. Never divide by a variable trig function, or you will lose valid solutions! $2\sin x \cos x - \cos x = 0$ $\cos x(2\sin x - 1) = 0$
4. Set each factor to zero.
   • $\cos x = 0 \implies x = \frac{\pi}{2}, \frac{3\pi}{2}$
   • $2\sin x - 1 = 0 \implies \sin x = \frac{1}{2} \implies x = \frac{\pi}{6}, \frac{5\pi}{6}$

The final solutions are $x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{3\pi}{2}$.

Key Strategies for Success

• Know your identities: Pythagorean, double-angle, and reciprocal identities are your primary tools for rewriting equations.
• Factor, don't divide: If you have $\sin x$ on both sides, subtract it to one side and factor it out. Dividing by $\sin x$ throws away the solutions where $\sin x = 0$.
• Check your interval: Pay close attention to the domain (e.g., $[0, 2\pi)$ vs $(-\infty, \infty)$). If no interval is given, you must write a general solution by adding $+ 2\pi k$ or $+ \pi k$ to your answers.