Trigonometry / Unit 7: Trigonometric Equations
7.1 Solving Basic Equations on an Interval
Notes
To solve an equation like $\sin x = a$ on $[0, 2\pi)$, find the reference angle, then use the quadrants where the function has that sign.
- Sine positive: Quadrants I and II. Negative: III and IV.
- Cosine positive: Quadrants I and IV. Negative: II and III.
- Tangent positive: Quadrants I and III. Negative: II and IV.
- Example: $\displaystyle \sin x = \frac{1}{2}$ gives reference angle $\displaystyle \frac{\pi}{6}$, so $\displaystyle x = \frac{\pi}{6}$ or $\displaystyle x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.
Practice
Solve $\displaystyle \sin x = \frac{1}{2}$ on $[0, 2\pi)$.
Sine is positive in Quadrants I and II, with reference angle $\displaystyle \frac{\pi}{6}$.
Solve $\displaystyle \cos x = -\frac{\sqrt{2}}{2}$ on $[0, 2\pi)$.
Cosine is negative in Quadrants II and III, with reference angle $\displaystyle \frac{\pi}{4}$.
Solve $\displaystyle \tan x = \sqrt{3}$ on $[0, 2\pi)$.
Tangent is positive in Quadrants I and III, with reference angle $\displaystyle \frac{\pi}{3}$.
Ready for more? Take the Unit 7 test.