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Small Angle Approximation

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Small Angle Approximation

The small angle approximation is a trick that can be used to estimate the values of trigonometric functions for small angles measured in radians (small values of). The small angle approximation is used to make it easier to solve and operate with trigonometric functions.

Small angle approximation equations

There are three equations we can use for the small angle approximation: one for , one for , and one for .

Small angle approximation for sin

The assumption that can be better understood when we look at the graphs of y = x and y = sin x.

small angle approximation sin x studysmarterA graph showing the lines y=x (red) and y=sin x (blue)

Now, as you can see around x = 0, the graphs of y = x and y = sinx are very close to each other.

small angle approximation sin x studysmarter

A graph showing the lines y=x (red) and y=sin x (blue) around x=0

This is why, for very small angles, we can say that

Small angle approximation for cos

The approximation for cosine isn't quite as straightforward as for sin. The small angle approximation for cos is derived using the small angle approximation result that we got for sin, and a double angle formula. We use the double angle formula:

Now, if we say that So,

We know from our previous calculation that for a small value ofwe assume that:

Which, simplified, gives us the small angle approximation for cos:

Small angle approximation for tan

For the small angle approximation of tan, we use the same logic as for sin. Looking at the graphs of y = tanx and y = x,

small angle approximation tan x studysmarterA graph showing the lines y=x (red) and y=tan x (blue)

Again, for values close to x = 0, we see that the two functions are very close to each other:

Small angle approximation Approximation  tan x StudySmarter

A graph showing the lines y=x (red) and y=tan x (blue) around x=0

Hence, we assume that for small values of

How to solve questions using the small angle approximation

The formulas derived earlier can be used in questions and problems to make it easier and quicker to solve them. We will look at a few examples of how to apply this.

When is small, show that can be approximated by

To solve this question, we will need to use the small angle approximations for sin and cos: We can now substitute this into , which gives us . We can simplify this expression by multiplying top and bottom by 2: which simplifies to , as required by the question.

a) When x is small, show that tan (3x) cos (2x) can be approximated by

b) Hence approximate the value of tan (0.3) cos (0.2) to 3 sf

This question needs to be answered in two parts: a and b. Let's start by looking at how we would solve a). We will need to use the facts did tan x≈ x and Substituting this into tan (3x) cos (2x), we get: . Multiplying the bracket by 3x:

as required.

For part b, we have to find the tan value for 0.3 and cos for 0.2. We know the expressions for tan 3x and cos 2x, so: 3x = 0.3 and 2x = 0.2 gives us x = 0.1. Now we can plug in 0.1 into the expression we found earlier:

If the angle is given in degrees, you will need to convert it to radians first to use the small angle approximation. You can use the formula

Small Angle Approximation - Key takeaways

  • The small angle approximation can be used to make it easier to work with trigonometric functions when looking at angles close to 0 rad.

  • The small angle approximation has to be worked out in radians.

  • The three formulas for small angle approximation are:

Frequently Asked Questions about Small Angle Approximation

No, small angle approximations have to be calculated in radians. If the angle is in degrees, you can convert the angle to radians first using radian=degree x (π/180)

The small angle approximation is a trick that we can use to make it easier to work with trigonometric functions when looking at small angles. The three small angle approximations are:

sin𝛉≈𝛉

cos𝛉≈1-(𝛉^2)/2

tan𝛉≈𝛉

To use the small angle approximation, simply replace sin𝛉 by 𝛉, cos 𝛉 by 1-(𝛉^2)/2 or tan𝛉 by 𝛉 in the question or problem that you are trying to solve and then carry on working it out as normal.

Final Small Angle Approximation Quiz

Question

 What unit does the angle need to be measured in for the small angle approximation to work?

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Answer

In radians and degrees if converted

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Question

What is the small angle approximation for sin 10°?

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Answer

0.1745

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Question

What is the small angle approximation for cos 10°?

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Answer

0.9848

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Question

Using small angle approximation, give the tan value for 5°

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Answer

0.0873

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Question

A function machine takes in two small angle approximations and multiplies them together. Jack put in sin(9°) and cos(9°). Jill puts in sin(8°) and tan(11°). Who ends up with the largest answer?

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Answer

Jack

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Question

Your manager wants to save time but be accurate. You are allowed a 2% error in your approximations otherwise you must find the precise value. For sin x, what integer angles in degrees would you not be allowed to approximate? Write your answer as an inequality.

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Answer

x>13°

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