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 . Let’s start with and compare it to line .
We can see that for small input values of , the graphs are very close; this tells us that for small input values of measured in radians. There are a few things worth noting about this approximation.
First, the phrase “small 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 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 , we can see that . This means that our approximation overestimates the value when is positive. Similarly, when , this approximation underestimates the value since we can see that .
Fourth, this approximation does not work if 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 being measured in radians.
We can find a similar approximation for the graph of by comparing it to .
Once again, we can see that for small input values of , the graphs are very close. This tells us that for small input values of measured in radians.
The approximation is more accurate the closer the values of are to 0, and we can see that if , then . Similarly, if , then .
Finding an approximation of is slightly more difficult since its graph does not look linear around . This means that we can try approximating the graph with a parabola as shown.
We can see that for small input values of , the graphs of and are very close; this tells us that for small input values of measured in radians. As before, this approximation is more accurate the closer the values of are to 0, and we can see that this approximation is always an underestimate: , for any value of .
We can write these approximations formally as follows.
Property: Small-Angle Approximations of Trigonometric Functions
- For a small angle measured in
radians,
we can approximate the outputs of the trigonometric function as follows:
- ,
- ,
- .
We can also note that we have equality in each of the approximations when .
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 for small values of measured in radians?
Answer
We first recall that we can approximate the trigonometric functions for small angles measured in radians using
We can note that if is a small angle measured in radians, then is also a small angle measured in radians. So,
Substituting these approximations into the given expression yields
Hence, for small values of measured in radians, we have
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 is a small angle measured in radians, , state the approximate value of for small values of measured in radians.
Answer
We first need to verify that for a small angle measured in radians, . We can do this by recalling the small-angle approximation of each of the trigonometric functions. We have for any small angle measured in radians.
We can note that if is a small angle measured in radians, then and are also small angles measured in radians. So, we can apply these approximations for these angles as well. We can substitute , , and into the three approximations, respectively, to get
We can now substitute these approximations into the given expression, giving us
Hence,
We can now note that since is a small angle measured in radians, and . This means that
Hence,
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 , given that 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
We note that if is a small angle measured in radians, then is also a small angle measured in radians. Therefore, these approximations hold when the angle is . Substituting into the approximations gives us
We can now substitute these approximations into the given expression and simplify to get
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 and then approximating using the approximation for for an angle measured in radians, calculate the percentage error in the approximation for to two significant figures.
Answer
We first recall that the percentage error in the approximation will be given by
We can calculate the exact value of using a calculator set to radians mode. We have .
We can approximate using the fact that for a small angle measured in radians we have
Thus,
We can substitute these values into the formula for the percentage error. We get
Hence, to two significant figures, the percentage error in this approximation is .
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 for a small positive angle measured in radians is , which of the following equations is true?
Answer
We first recall that the percentage error in the approximation will be given by
Our approximation is , and we are told that the percentage error is . Thus, we have the following equation:
We can divide the equation through by 100 and simplify to get
Thus, either or .
Since is a positive small angle measured in radians, we can recall that . So, . This means that we cannot have . Thus, the only solution to this equation is
We can rearrange to get
This is choice A.
Let’s finish by recapping some of the important points from this explainer.
Key Points
- For a small angle measured in
radians,
we can approximate the outputs of the trigonometric function as follows:
- ,
- ,
- .
- The approximation for is an overestimate when and an underestimate when . The approximation for is an underestimate when and an overestimate when . The approximation for is always an underestimate: .
- We can calculate the percentage error in these approximations to see how accurate the approximations are. We have
