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Small angle approximations

Small angle approximations

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Summary

Small angle approximations

​​In a nutshell

Small angle approximations involve approximating the values of sin⁡θ\sin \thetasinθ, cos⁡θ\cos \thetacosθ or tan⁡θ\tan \thetatanθ to a linear or quadratic expression when the value of θ\thetaθ is small. Small angle approximations are useful as they remove the need for any trigonometric functions and can therefore simplify complex calculations. The approximations only apply if the angle θ\thetaθ is in radians.



Small angle approximations

When θ\thetaθ is small and in radians, the following approximations can be used. The graphs illustrate how close the trigonometric function is to the approximation. 

​

Maths; Radians; KS5 Year 13; Small angle approximations
Maths; Radians; KS5 Year 13; Small angle approximations
Maths; Radians; KS5 Year 13; Small angle approximations
​sin⁡θ≈θ\boxed{\sin \theta \approx \theta}sinθ≈θ​​​
​cos⁡θ≈1−θ22\boxed{\cos \theta \approx 1-\dfrac{\theta^2}{2}}cosθ≈1−2θ2​​​​
​tan⁡θ≈θ\boxed{\tan \theta \approx \theta}tanθ≈θ​​​

​

Example 1

Find an approximation for f(x)f(x)f(x) when the angle xxx is small, but non-zero. 

f(x)=sin⁡(x)tan⁡(x)1−cos⁡(3x)f(x) = \dfrac {\sin(x) \tan (x)}{1 - \cos (3x)}f(x)=1−cos(3x)sin(x)tan(x)​​​


Substitute in the approximations and simplify:

f(x)=sin⁡(x)tan⁡(x)1−cos⁡(3x)≈x×x1−(1−(3x)22)≈x21−1+9x22≈x29x22≈29‾\begin{aligned}f(x) &= \dfrac {\sin(x) \tan (x)}{1 - \cos (3x)} \\\\&\approx \dfrac{x \times x}{1 - (1-\frac{(3x)^2}{2})} \\\\&\approx \dfrac{x ^2}{1 - 1+\frac{9x^2}{2}} \\\\&\approx \dfrac{x ^2}{\frac{9x^2}{2}} \\\\&\approx \underline{ \dfrac{2}{9}} \\\\\end{aligned}f(x)​=1−cos(3x)sin(x)tan(x)​≈1−(1−2(3x)2​)x×x​≈1−1+29x2​x2​≈29x2​x2​≈92​​​​​



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FAQs - Frequently Asked Questions

When can you use small angle approximations?

You can use the small angle approximations when the angle is small and in radians.

What are small angle approximations?

Small angle approximations involve approximating the values of sin(x), cos(x) or tan(x) to a linear or quadratic expression when the value of x is small.

Why are small angle approximations useful?

Small angle approximations are useful as they remove the need for any trigonometric functions and can therefore simplify complex calculations.

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