Small Angle Approximations
Small Angle Approximations
- When \(\theta\) is small:
- \(\sin\theta=\theta\)
- \(\tan\theta=\theta\)
- \(\cos\theta=1-\frac{\theta^2}{2}\)
\(\frac{\sin4\theta-\tan2\theta}{3\theta}\)
- \(=\frac{4\theta-2\theta}{3\theta}\)
- \(=\frac{2\theta}{3\theta}\)
- \(=\frac{2}{3}\)
\(\frac{1-\cos2\theta}{\tan2\theta \sin\theta}\)
- \(=\frac{1-\left( 1-\frac{4\theta^2}{2} \right)}{2\theta \times\theta}\)
- \(=\frac{2\theta^2}{2\theta^2}=1\)
\(\frac{3\tan\theta-\theta}{\sin2\theta}\)
- \(=\frac{3\theta-\theta}{2\theta}\)
- \(=\frac{2\theta}{2\theta}\)
- \(=2\)
\(\frac{\tan4\theta+\theta^2}{3\theta-2\sin\theta}\)
- \(=\frac{4\theta+\theta^2}{3\theta-2\theta}\)
- \(=\frac{4\theta+\theta^2}{\theta}\)
- \(=4+\theta\)
Question 3
- \(\cos(0.244)=0.970379\)
- \(1-\frac{0.244^2}{2}=0.970232\)
- \(0.015\%\)
- \(1.768\)
- The answer to C is more accurate because the angle is smaller and small angle approximations get more accurate as the angle gets smaller.
\(\frac{4\cos3\theta-2+5\sin\theta}{1-\sin2\theta}\)
- \(\frac{4\cos3\theta-2+5\sin\theta}{1-\sin2\theta}\)
- \(=\frac{4\left( 1-\frac{9\theta^2}{2} \right)-2+5\theta}{1-2\theta}\)
- \(=\frac{4-18\theta^2-2+5\theta}{1-2\theta}\)
- \(=\frac{2-18\theta^2+5\theta}{1-2\theta}\)
- \(\frac{\cancel{ (1-2\theta) }(9\theta+2)}{\cancel{ 1-2\theta }}\)
- \(=9\theta+2\)
When \(\theta\) is small, \(\frac{4\cos3\theta-2+5\sin\theta}{1-\sin2\theta}\approx2\)