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Lesson Explainer: Small-Angle Approximations Mathematics

In this explainer, we will learn how to approximate trigonometric functions when the angle of the function is close to zero.

Finding the exact value of the outputs of the trigonometric functions without using a calculator is a very difficult task. Instead, we often use approximations to make it easier to perform difficult calculations. Of course this has its drawbacks, such as the answers not being as accurate.

We can find approximations of the trigonometric functions for small angles measured in radians by considering their graphs near input values of π‘₯=0. Let’s start with 𝑦=π‘₯sin and compare it to line 𝑦=π‘₯.

We can see that for small input values of π‘₯, the graphs are very close; this tells us that sinπ‘₯β‰ˆπ‘₯ for small input values of π‘₯ measured in radians. There are a few things worth noting about this approximation.

First, the phrase β€œsmall input values” is not very rigorous since we are not describing exactly how small the values need to be for this approximation to be valid. How small the inputs need to be would depend on how accurate we want the approximation to be.

Second, we can see that as the values of π‘₯ get larger, this approximation gets worse. In fact, the opposite is also true; as the input values get closer to 0, the approximation becomes more accurate.

Third, if π‘₯>0, we can see that π‘₯>π‘₯sin. This means that our approximation overestimates the value when π‘₯ is positive. Similarly, when π‘₯<0, this approximation underestimates the value since we can see that π‘₯<π‘₯sin.

Fourth, this approximation does not work if π‘₯ is measured in degrees (or any unit other than radians). This is because the shape of the sine function will change depending on how we measure the inputs; this means that we would need a different approximation for these graphs. We will not go into detail for these approximations since we are only interested in π‘₯ being measured in radians.

We can find a similar approximation for the graph of 𝑦=π‘₯tan by comparing it to 𝑦=π‘₯.

Once again, we can see that for small input values of π‘₯, the graphs are very close. This tells us that tanπ‘₯β‰ˆπ‘₯ for small input values of π‘₯ measured in radians.

The approximation is more accurate the closer the values of π‘₯ are to 0, and we can see that if π‘₯>0, then tanπ‘₯>π‘₯. Similarly, if π‘₯<0, then tanπ‘₯<π‘₯.

Finding an approximation of cosπ‘₯ is slightly more difficult since its graph does not look linear around π‘₯=0. This means that we can try approximating the graph with a parabola as shown.

We can see that for small input values of π‘₯, the graphs of cosπ‘₯ and 1βˆ’π‘₯2 are very close; this tells us that cosπ‘₯β‰ˆ1βˆ’π‘₯2 for small input values of π‘₯ measured in radians. As before, this approximation is more accurate the closer the values of π‘₯ are to 0, and we can see that this approximation is always an underestimate: 1βˆ’π‘₯2≀π‘₯cos, for any value of π‘₯.

We can write these approximations formally as follows.

Property: Small-Angle Approximations of Trigonometric Functions

  • For a small angle π‘₯ measured in radians, we can approximate the outputs of the trigonometric function as follows:
    • sinπ‘₯β‰ˆπ‘₯,
    • cosπ‘₯β‰ˆ1βˆ’π‘₯2,
    • tanπ‘₯β‰ˆπ‘₯.

We can also note that we have equality in each of the approximations when π‘₯=0.

In our first example, we will use the approximations to find an approximate value of a trigonometric expression.

Example 1: Approximating a Trigonometric Expression Using the Small-Angle Approximations

What is the approximate value of sintantan2π‘₯βˆ’π‘₯+π‘₯2π‘₯ for small values of π‘₯ measured in radians?

Answer

We first recall that we can approximate the trigonometric functions for small angles measured in radians using sintanπ‘₯β‰ˆπ‘₯,π‘₯β‰ˆπ‘₯.

We can note that if π‘₯ is a small angle measured in radians, then 2π‘₯ is also a small angle measured in radians. So, sintantan2π‘₯β‰ˆ2π‘₯,π‘₯β‰ˆπ‘₯,2π‘₯β‰ˆ2π‘₯.

Substituting these approximations into the given expression yields sintantan2π‘₯βˆ’π‘₯+π‘₯2π‘₯β‰ˆ2π‘₯βˆ’π‘₯+π‘₯2π‘₯=2π‘₯2π‘₯=1.

Hence, for small values of π‘₯ measured in radians, we have sintantan2π‘₯βˆ’π‘₯+π‘₯2π‘₯β‰ˆ1.

In our next example, we will find an approximate value for a trigonometric expression using the small-angle approximations for the trigonometric functions.

Example 2: Using Small-Angle Approximations to Approximate a Trigonometric Expression

By first showing that when πœƒ is a small angle measured in radians, sincostan2πœƒβˆ’4πœƒ+πœƒβ‰ˆ8πœƒ+3πœƒβˆ’1, state the approximate value of sincostan2πœƒβˆ’4πœƒ+πœƒ for small values of πœƒ measured in radians.

Answer

We first need to verify that for a small angle πœƒ measured in radians, sincostan2πœƒβˆ’4πœƒ+πœƒβ‰ˆ8πœƒ+3πœƒβˆ’1. We can do this by recalling the small-angle approximation of each of the trigonometric functions. We have sincostanπ‘₯β‰ˆπ‘₯,π‘₯β‰ˆ1βˆ’π‘₯2,π‘₯β‰ˆπ‘₯. for any small angle π‘₯ measured in radians.

We can note that if πœƒ is a small angle measured in radians, then 2πœƒ and 4πœƒ are also small angles measured in radians. So, we can apply these approximations for these angles as well. We can substitute π‘₯=2πœƒ, π‘₯=4πœƒ, and π‘₯=πœƒ into the three approximations, respectively, to get sincostan2πœƒβ‰ˆ2πœƒ,4πœƒβ‰ˆ1βˆ’(4πœƒ)2=1βˆ’8πœƒ,πœƒβ‰ˆπœƒ.

We can now substitute these approximations into the given expression, giving us sincostan2πœƒβˆ’4πœƒ+πœƒβ‰ˆ2πœƒβˆ’ο€Ή1βˆ’8πœƒο…+πœƒ=8πœƒ+3πœƒβˆ’1.

Hence, sincostan2πœƒβˆ’4πœƒ+πœƒβ‰ˆ8πœƒ+3πœƒβˆ’1.

We can now note that since πœƒ is a small angle measured in radians, 8πœƒβ‰ˆ0 and 3πœƒβ‰ˆ0. This means that 8πœƒ+3πœƒβˆ’1β‰ˆβˆ’1.

Hence, sincostan2πœƒβˆ’4πœƒ+πœƒβ‰ˆβˆ’1.

Let’s now see an example of finding an approximate expression for the value of a trigonometric expression using the small-angle approximations.

Example 3: Using Small-Angle Approximations to Approximate a Trigonometric Expression

Find an expression for the approximate value of sintancos2π‘₯+π‘₯+12π‘₯+2π‘₯, given that π‘₯ is a small angle measured in radians.

Answer

We first recall that we can approximate the trigonometric functions for small angles measured in radians using sincostanπ‘₯β‰ˆπ‘₯,π‘₯β‰ˆ1βˆ’π‘₯2,π‘₯β‰ˆπ‘₯.

We note that if π‘₯ is a small angle measured in radians, then 2π‘₯ is also a small angle measured in radians. Therefore, these approximations hold when the angle is 2π‘₯. Substituting 2π‘₯ into the approximations gives us sincostan2π‘₯β‰ˆ2π‘₯,2π‘₯β‰ˆ1βˆ’(2π‘₯)2=1βˆ’2π‘₯,π‘₯β‰ˆπ‘₯.

We can now substitute these approximations into the given expression and simplify to get sintancos2π‘₯+π‘₯+12π‘₯+2π‘₯β‰ˆ2π‘₯+π‘₯+11βˆ’2π‘₯+2π‘₯=3π‘₯+11=3π‘₯+1.

In our next example, we will calculate the percentage error in an approximation using the small-angle approximations.

Example 4: Finding and Understanding the Error When Using Small-Angle Approximations

By first calculating cos(0.35) and then approximating cos(0.35) using the approximation for cos(πœƒ) for an angle measured in radians, calculate the percentage error in the approximation for cos(0.35) to two significant figures.

Answer

We first recall that the percentage error in the approximation will be given by percentageerrorexactvalueapproximatedvalueexactvalue=|||βˆ’|||Γ—100.

We can calculate the exact value of cos(0.35) using a calculator set to radians mode. We have cos(0.35)=0.939….

We can approximate cos(0.35) using the fact that for a small angle π‘₯ measured in radians we have cosπœƒβ‰ˆ1βˆ’πœƒ2.

Thus, cos(0.35)β‰ˆ1βˆ’(0.35)2=0.93875.

We can substitute these values into the formula for the percentage error. We get percentageerrorcoscos=|||(0.35)βˆ’0.93875(0.35)|||Γ—100=0.0662….

Hence, to two significant figures, the percentage error in this approximation is 0.066%.

In our final example, we will use the given percentage error in a small-angle approximation to determine whether a statement is true.

Example 5: Using a Percentage Error for a Small-Angle Approximation to Prove a Statement

Given that the percentage error for an approximation of tanπ‘₯ for a small positive angle π‘₯ measured in radians is 2%, which of the following equations is true?

  1. 49π‘₯β‰ˆ50π‘₯tan
  2. 50π‘₯β‰ˆ49π‘₯tan
  3. 51π‘₯β‰ˆ49π‘₯tan
  4. 50π‘₯β‰ˆ51π‘₯tan
  5. 48π‘₯β‰ˆ51π‘₯tan

Answer

We first recall that the percentage error in the approximation will be given by percentageerrorexactvalueapproximatedvalueexactvalue=|||βˆ’|||Γ—100.

Our approximation is tanπ‘₯β‰ˆπ‘₯, and we are told that the percentage error is 2%. Thus, we have the following equation: 2β‰ˆ|||(π‘₯)βˆ’π‘₯(π‘₯)|||Γ—100.tantan

We can divide the equation through by 100 and simplify to get 2100β‰ˆ||1βˆ’π‘₯π‘₯||150β‰ˆ||1βˆ’π‘₯π‘₯||.tantan

Thus, either π‘₯π‘₯β‰ˆ5150tan or π‘₯π‘₯β‰ˆ4950tan.

Since π‘₯ is a positive small angle measured in radians, we can recall that tanπ‘₯>π‘₯. So, π‘₯π‘₯<1tan. This means that we cannot have π‘₯π‘₯β‰ˆ5150tan. Thus, the only solution to this equation is π‘₯π‘₯β‰ˆ4950.tan

We can rearrange to get 49π‘₯β‰ˆ50π‘₯.tan

This is choice A.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • For a small angle π‘₯ measured in radians, we can approximate the outputs of the trigonometric function as follows:
    • sinπ‘₯β‰ˆπ‘₯,
    • cosπ‘₯β‰ˆ1βˆ’π‘₯2,
    • tanπ‘₯β‰ˆπ‘₯.
  • The approximation for sinπ‘₯ is an overestimate when π‘₯>0 and an underestimate when π‘₯<0. The approximation for tanπ‘₯ is an underestimate when π‘₯>0 and an overestimate when π‘₯<0. The approximation for cosπ‘₯ is always an underestimate: 1βˆ’π‘₯2≀π‘₯cos.
  • We can calculate the percentage error in these approximations to see how accurate the approximations are. We have percentageerrorexactvalueapproximationexactvalue=|||βˆ’|||Γ—100.

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