Small Angle Approximations

Expert reviewed • 21 July 2024 • 3 minute read

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What are Small Angle Approximations?

Small angle approximations are useful simplifications which make certain trigonometric calculations easier, specifically when dealing with angles close to zero.

Thus, when an angle is expressed in radians for a value of $x$ that is considered "small", we can use the following formulas to approximate the trigonometric functions:

sinx\approx x\qquad\quad cosx\approx 1\qquad\quad tanx\approx x

However, while small angle approximations are useful for simplifying calculations, it's important to note their limitations for potential error in situations where higher precision is needed, or the provided angle is not sufficiently small.

Practice Question 1

Calculate the small angle approximation of $tan(0.5\degree)$

From this question, we must identify that the small angle approximation formula $tanx\approx x$ must be used.

However, we first need to convert $0.5\degree$ to radians. This is done by multiplying the degrees by $\frac{\pi}{180}$

0.5\degree \times \frac{\pi}{180} = \frac{\pi}{360}

Now, substituting into the small approximation formula, we find that:

tan(0.5\degree)\approx \frac{\pi}{360}

Up Next

Differentiating Trigonometric Functions

Return to Module 6: The Trigonometric Functions