Trigonometry / Unit 7: Trigonometric Equations
7.3 General Solutions
Notes
Trig functions repeat, so on all real numbers there are infinitely many solutions. Add the period times an integer $n$.
- $\sin x = a$: $x = \alpha + 2\pi n$ or $x = \pi - \alpha + 2\pi n$.
- $\cos x = a$: $x = \pm\alpha + 2\pi n$.
- $\tan x = a$: $x = \alpha + \pi n$ (tangent has period $\pi$).
- Example: $\displaystyle \sin x = \frac{\sqrt{3}}{2}$ gives $\displaystyle x = \frac{\pi}{3} + 2\pi n$ or $\displaystyle x = \frac{2\pi}{3} + 2\pi n$.
Practice
What is the general solution of $\tan x = 1$?
Tangent has period $\pi$, so the solutions are $\displaystyle \frac{\pi}{4}$ plus multiples of $\pi$.
What is the general solution of $\displaystyle \cos x = \frac{1}{2}$?
Cosine is $\displaystyle \frac{1}{2}$ at $\displaystyle \frac{\pi}{3}$ and $\displaystyle -\frac{\pi}{3}$ (that is, $\displaystyle \frac{5\pi}{3}$), then repeats every $2\pi$.
What is the general solution of $\sin x = -1$?
Sine equals $-1$ only at $\displaystyle \frac{3\pi}{2}$ in each cycle.
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