5.4.3 Small Angle Approximations

Small angle approximations are useful techniques in trigonometry and calculus for simplifying the values of trigonometric functions when the angle $\theta$ is small (typically measured in radians). These approximations are particularly helpful when dealing with limits, series expansions, or problems where the angle is close to zero.

1. Key Small Angle Approximations:

For very small angles $\theta$ (in radians), the following approximations hold:

• Sine:

$\sin \theta \approx \theta$

• Cosine:

$\cos \theta \approx 1 - \frac{\theta^2}{2}$

• Tangent:

$\tan \theta \approx \theta$

2. Understanding the Approximations:

• Sine Approximation: $\sin \theta \approx \theta$ As $\theta$ approaches 0, the sine of the angle becomes nearly equal to the angle itself when $\theta$ is in radians. This is because the graph of $\sin \theta$ is very close to the line y = $\theta$ for small values of $\theta$.

• Cosine Approximation: $\cos \theta \approx 1 - \frac{\theta^2}{2}$ For small angles, $\cos \theta$ remains close to 1, with the first correction term being $-\frac{\theta^2}{2}$. The cosine graph is nearly flat near $\theta = 0,$ which is why the approximation starts with 1.

• Tangent Approximation: $\tan \theta \approx \theta$ Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, and both $\sin \theta \approx \theta$ and $\cos \theta \approx 1$ for small $\theta$, the tangent function also approximates to the angle itself.

3. Derivation Using Taylor Series:

These approximations can be derived from the Taylor series expansions of the trigonometric functions around $\theta = 0:$

• Sine:

$\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots$

For small $\theta$, higher-order terms like $\frac{\theta^3}{6}$ become negligible, so:

$\sin \theta \approx \theta$

• Cosine:

$\cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots$

For small $\theta$, higher-order terms like $\frac{\theta^4}{24}$ become negligible, so:

$\cos \theta \approx 1 - \frac{\theta^2}{2}$

• Tangent:

$\tan \theta = \frac{\sin \theta}{\cos \theta} \approx \frac{\theta}{1} = \theta$

4. Applications of Small Angle Approximations:

• Physics: Small angle approximations are frequently used in physics, particularly in pendulum motion, where the sine of the angle of displacement is approximated as the angle itself for small oscillations.
• Engineering: In engineering, small angle approximations simplify the analysis of structures and mechanisms where small angular displacements occur.
• Astronomy: The approximation is also used in astronomy for calculating the angular diameter of celestial objects when viewed from Earth.

5. Example Problems Using Small Angle Approximations:

Summary:

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Small Angle Approximations

When dealing with small angles measured in radians, there are certain useful approximations we can use if the context of the problem allows approximations.

Graph: A graph is shown with two lines:

• The blue line represents $y = x$.
• The red curve represents $y = \sin(\theta)$.

The above graph suggests that for small $\theta, \sin(\theta) \approx \theta$.

Using $\sin(\theta) \approx \theta$, we can derive another approximation for $\cos(\theta)$:

$$\cos(\theta) = \cos\left(\frac{\theta}{2} + \frac{\theta}{2}\right) = 1 - 2\sin^2\left(\frac{\theta}{2}\right)$$

(Using double angle formula)

For small $\theta, \sin(\theta) \approx \theta \Rightarrow \cos(\theta) \approx 1 - 2\left(\frac{\theta}{2}\right)^2 = 1 - \frac{2\theta^2}{4}$

$$= 1 - \frac{\theta^2}{2}$$ $$\therefore \text{ For small } \theta, \cos(\theta) \approx 1 - \frac{\theta^2}{2}$$

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For small $\theta \quad \\\tan\theta \approx \theta$

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5. a) When $\theta$ is small, show that the expression $\dfrac{4 \cos 3\theta - 2 + 5 \sin \theta}{1 - \sin 2\theta}$

can be written as $9\theta + 2$.

b) Hence write down the value of $\dfrac{4 \cos 3\theta - 2 + 5 \sin \theta}{1 - \sin 2\theta}$ when $\theta$ is small.

When $\theta$ is small,

$$\cos 3\theta \approx 1 - \frac{(3\theta)^2}{2} = 1 - \frac{9\theta^2}{2}$$ $$\sin \theta \approx \theta, \quad \sin 2\theta \approx 2\theta$$ $$\frac{4(1 - \frac{9\theta^2}{2}) - 2 + 5\theta}{1 - 2\theta}$$ $$= \frac{4 - 18\theta^2 - 2 + 5\theta}{1 - 2\theta}$$ $$\Rightarrow -\frac{18\theta^2 - 5\theta - 2}{1 - 2\theta} = \frac {\cancel-(9\theta+2)\cancel{(2\theta-1)}}{\cancel-\cancel{(2\theta-1)}}$$ $$\Rightarrow 9\theta +2$$

Assuming $\theta$ is small, we can approximate the expression as $9\theta + 2$.

Hence $\dfrac{4\cos 3\theta - 2 + 5\sin \theta}{1 - \sin 2\theta} \approx 9\theta + 2$ when $\theta$ is small.

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