Chapter 5 - Trigonometric Equations
| Sunday, 29 August 2021 | |
| 2-minute read | |
| 331 words | |
5.3 Solving Trigonometric Equations
- Solve \(2\sin x - 1\)
\[ 2\sin x = 1 \]
\[ \sin x = \frac{1}{2} \]
\[ x = \frac{\pi}{6} + 2\pi k\]
- Solve \(\sin x + \sqrt{2} = -\sin x\)
\[ 2sin x = -\sqrt{2} \]
\[ sin x = -\frac{\sqrt{2}}{2} \]
\[ x = -\frac{\pi}{3} + 2\pi k \]
- Solve \(3\tan^2 x - 1 = 0\)
\[ \tan^2 x = \frac{1}{3} \]
\[ \tan x = \pm \frac{1}{\frac{3}} \]
Don't forget the plus/minus sign.
\[ x = \frac{\pi k}{6}, \textrm{for} k = 1, 5, 7, 11 \]
- Solve \(\cot x \cos^2 x = 2 \cot x\)
\[ \cos^2 x = \cot x \]
\[ \cot x \cos^2 x - 2 \cot x = 0 \]
\[ \cot x(\cos^2 x - 2) = 0 \]
\[ \cot x = 0, \cos^2x - 2 = 0 \]
\[ \cot x = 0, x = { \frac{\pi}{2}, \frac{2\pi}{2} } \]
\[ \cos^2x - 2 = 0, x = \emptyset \]
\[ x = { \frac{\pi}{2}, \frac{2\pi}{2} } \]
- Solve \(2\sin^2 x - \sin x - 1 = 0\)
\[ (\sin x - 1)(2\sin x + 1) \]
\[ \sin x = 1, x = \frac{\pi}{2} \]
\[ \sin x = -\frac{1}{2}, x = { \frac{7\pi}{6}, \frac{11\pi}{6} } \]
\[ x = { \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} } \]
5.4 Sum and Difference Identities
Sum and Difference Identities
\[ \sin(x \pm y) = \sin x \cos y \pm \cos x \sin y \] \[ \cos(x \pm y) = \cos x \cos y \mp \sin x \sin y \] \[ \tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y} \]
Example
- Evaluate \(\cos75°\)
\[ \cos75° = \cos(120°-45°) = \cos120°\cos45° + sin120°sin45° \]
5.5 Multiple and Half Angle Identities
\[ \sin(2\theta) = \sin(\theta + \theta) \]
\[ = \sin\theta\cos\theta + \cos\theta\sin\theta \]
\[ = 2\sin\theta\cos\theta \]
\[ \sin(3\theta) = \sin(2\theta + \theta) = (2\sin\theta\cos\theta)\cos(\theta) + (1 - 2\sin^2\theta)\sin(\theta) \]
$$ sin(\frac{θ}{2}) = ± \sqrt{\frac{1 - cosθ}{2}}