Small Angle Approximations

Expert reviewed • 22 November 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 xx that is considered "small", we can use the following formulas to approximate the trigonometric functions:

sinx≈xcosx≈1tanx≈xsinx\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°)tan(0.5\degree)

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

However, we first need to convert 0.5°0.5\degree to radians. This is done by multiplying the degrees by π180\frac{\pi}{180}

0.5°×π180=π3600.5\degree \times \frac{\pi}{180} = \frac{\pi}{360}

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

tan(0.5°)≈π360tan(0.5\degree)\approx \frac{\pi}{360}

Up Next

Differentiating Trigonometric Functions

Return to Module 6: The Trigonometric Functions