Trigonometry / Unit 7: Trigonometric Equations
7.4 Factoring and Quadratic-Type Equations
Notes
Treat $\sin x$ or $\cos x$ like a variable and factor, just as in algebra.
- Move everything to one side, factor, and set each factor to 0. Never divide by a trig function that could be zero, since that loses solutions.
- Example: $2\sin^2 x - \sin x = 0 \Rightarrow \sin x (2\sin x - 1) = 0$, so $\sin x = 0$ or $\displaystyle \sin x = \frac{1}{2}$.
- If a factor gives a value outside $[-1, 1]$ for sine or cosine, it has no solutions.
Practice
Solve $2\cos^2 x - \cos x = 0$ on $[0, 2\pi)$.
$\cos x (2\cos x - 1) = 0$, so $\cos x = 0$ or $\displaystyle \cos x = \frac{1}{2}$.
Solve $2\sin^2 x + \sin x - 1 = 0$ on $[0, 2\pi)$.
$(2\sin x - 1)(\sin x + 1) = 0$ gives $\displaystyle \sin x = \frac{1}{2}$ or $\sin x = -1$.
Solve $\tan^2 x = 3$ on $[0, \pi)$.
$\displaystyle \tan x = \pm\sqrt{3}$: $\displaystyle \tan x = \sqrt{3}$ at $\displaystyle \frac{\pi}{3}$ and $\displaystyle \tan x = -\sqrt{3}$ at $\displaystyle \frac{2\pi}{3}$.
Ready for more? Take the Unit 7 test.