Precalculus / Unit 5: Trigonometric Identities and Equations
5.5 Solving Basic Trigonometric Equations
Notes
Solving a trig equation means finding all angles that make it true.
- Isolate the trig function, find the reference angle, then use the quadrants.
- On $[0, 2\pi)$ there are usually two solutions for $\sin x = k$ or $\cos x = k$ (with $|k| < 1$), and two for $\tan x = k$.
- Factor when you can. Example: $2\cos^2 x - \cos x - 1 = 0 \Rightarrow (2\cos x + 1)(\cos x - 1) = 0$.
- Example: $\tan x = -1$ on $[0, 2\pi)$ gives $\displaystyle x = \frac{3\pi}{4}, \frac{7\pi}{4}$.
Practice
Solve $2\sin x - 1 = 0$ on $[0, 2\pi)$.
$\displaystyle \sin x = \frac{1}{2}$ in quadrants I and II.
Solve $2\cos^2 x - \cos x - 1 = 0$ on $[0, 2\pi)$.
$\displaystyle \cos x = -\frac{1}{2}$ gives $\displaystyle \frac{2\pi}{3}, \frac{4\pi}{3}$ and $\cos x = 1$ gives $0$.
Solve $\tan x = -1$ on $[0, 2\pi)$.
Tangent is negative in quadrants II and IV with reference angle $\displaystyle \frac{\pi}{4}$.
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