3.6 Inverse Trigonometric Functions

Learning Objectives

In this section, you will:

  • Understand and use the inverse sine, cosine, and tangent functions.
  • Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
  • Use a calculator to evaluate inverse trigonometric functions.
  • Find exact values of composite functions with inverse trigonometric functions.

For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions.

Understanding and Using the Inverse Sine, Cosine, and Tangent Functions

In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure 1.

A chart that says “Trig Functinos”, “Inverse Trig Functions”, “Domain: Measure of an angle”, “Domain: Ratio”, “Range: Ratio”, and “Range: Measure of an angle”.

Figure 1

For example, if f(x)=sin(x),f(x)=sin(x), then we would write f1(x)=sin1(x). Be aware that sin1(x) does not mean 1sin(x).  The following examples illustrate the inverse trigonometric functions:

  • Since sin(π6)=12, then π6=sin1(12).
  • Since cos(π)=1, then π=cos1(1).
  • Since tan(π4)=1, then π4=tan1(1).

In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if f(a)=b, then an inverse function would satisfy f1(b)=a.

Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0. Figure 2 shows the graph of the sine function limited to [π2,π2] and the graph of the cosine function limited to [0,π].

Two side-by-side graphs. The first graph, graph A, shows half of a period of the function sine of x. The second graph, graph B, shows half a period of the function cosine of x.

Figure 2: (a) Sine function on a restricted domain of [π2,π2]; (b) Cosine function on a restricted domain of [0,π]

Figure 3 shows the graph of the tangent function limited to (π2,π2).

A graph of one period of tangent of x, from -pi/2 to pi/2.

Figure 3: Tangent function on a restricted domain of (π2,π2)

These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.

On these restricted domains, we can define the inverse trigonometric functions.

  • The inverse sine function y=sin1(x) means x=sin(y). The inverse sine function is sometimes called the arcsine function, and notated arcsin(x).
    y=sin1(x) has domain [1,1] and range [π2,π2].
  • The inverse cosine function y=cos1(x) means x=cos(y). The inverse cosine function is sometimes called the arccosine function, and notated arccos(x).
    y=cos1(x) has domain [1,1] and range [0,π].
  • The inverse tangent function y=tan1(x) means x=tan(y). The inverse tangent function is sometimes called the arctangent function, and notated arctan(x).
    y=tan1(x) has domain (,) and range (π2,π2).

The graphs of the inverse functions are shown in Figure 4, Figure 5, and Figure 6. Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that sin1(x) has domain [1,1] and range [π2,π2], cos1(x) has domain [1,1] and range [0,π], and tan1(x) has domain of all real numbers and range (π2,π2). To find the domain and range of inverse trigonometric functions, switch the restricted domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line y=x.

A graph of the functions of sine of x and arc sine of x. There is a dotted line y=x between the two graphs, to show inverse nature of the two functions

Figure 4: The sine function and inverse sine (or arcsine) function.

A graph of the functions of cosine of x and arc cosine of x. There is a dotted line at y=x to show the inverse nature of the two functions.

Figure 5: The cosine function and inverse cosine (or arccosine) function.

A graph of the functions of tangent of x and arc tangent of x. There is a dotted line at y=x to show the inverse nature of the two functions.

Figure 6: The tangent function and inverse tangent (or arctangent) function.

Relations for Inverse Sine, Cosine, and Tangent Functions

For angles in the interval [π2,π2], if sin(x)=y, then sin1(y)=x.

For angles in the interval [0,π], if cos(x)=y, then cos1(y)=x.

For angles in the interval (π2,π2), if tan(x)=y, then tan1(y)=x.

Example 1: Writing a Relation for an Inverse Function

Given sin(5π12)0.96593, write a relation involving the inverse sine.

Try It #1

Given cos(0.5)0.8776, write a relation involving the inverse cosine.

Finding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions

Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically π6 (30°), π4 (45°), and π3 (60°), and the reflections into other quadrants.

How To

Given a “special” input value, evaluate an inverse trigonometric function.

  1. Find angle x for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.
  2. If x is not in the defined range of the inverse, find another angle that is in the defined range and has the same sine, cosine, or tangent as x, depending on which corresponds to the given inverse function.

Example 2: Evaluating Inverse Trigonometric Functions for Special Input Values

Evaluate each of the following.

  1. sin1(12)
  2. sin1(22)
  3. cos1(32)
  4. tan1(1)

Try It #2

Evaluate each of the following.

  1. sin1(1)
  2. tan1(1)
  3. cos1(1)
  4. cos1(12)

Using a Calculator to Evaluate Inverse Trigonometric Functions

To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example, SIN 1, ARCSIN, or ASIN.

In Section 3.1, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places.

In these examples and exercises, the answers will be interpreted as angles and we will use θ as the independent variable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to the application.

Example 3: Evaluating the Inverse Sine on a Calculator

Evaluate sin1(0.97) using a calculator.

Try It #3

Evaluate cos1(0.4) using a calculator.

How To

Given two sides of a right triangle like the one shown in  Figure 7, find an angle.

An illustration of a right triangle with an angle theta. Adjacent to theta is the side a, opposite theta is the side p, and the hypoteneuse is side h.

Figure 7

  1. If one given side is the hypotenuse of length h and the side of length a adjacent to the desired angle is given, use the equation  θ=cos1(ah).
  2. If one given side is the hypotenuse of length h and the side of length p opposite to the desired angle is given, use the equation θ=sin1(ph).
  3. If the two legs (the sides adjacent to the right angle) are given, then use the equation θ=tan1(pa).

Example 4: Applying an Inverse Function to a Right Triangle

Solve the triangle in Figure 8 for the angle θ.

An illustration of a right triangle with the angle theta. Adjacent to the angle theta is a side with a length of 9 and a hypoteneuse of length 12.

Figure 8

Try It #4

Solve the triangle in Figure 9 for the angle θ.

An illustration of a right triangle with the angle theta. Opposite to the angle theta is a side with a length of 6 and a hypoteneuse of length 10.

Figure 9

Finding Exact Values of Composite Functions with Inverse Trigonometric Functions

There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output.

Evaluating Compositions of the Form f(f1(y)) and f1(f(x))

For any trigonometric function, f(f1(y))=y for all y in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of f was defined to be identical to the domain of f1. However, we have to be a little more careful with expressions of the form f1(f(x)).

Compositions of a Trigonometric Function and its Inverse

sin(sin1(x))=x for 1x1cos(cos1(x))=x for 1x1tan(tan1(x))=x for <x<sin1(sin(x))=x only for π2xπ2cos1(cos(x))=x only for 0xπtan1(tan(x))=x only for π2<x<π2

Q&A

Is it correct that sin1(sin(x))=x?

No. This equation is correct if x belongs to the restricted domain [π2,π2], but sine is defined for all real input values, and for x outside the restricted interval, the equation is not correct because its inverse always returns a value in [π2,π2]. The situation is similar for cosine and tangent and their inverses. For example, sin1(sin(3π4))=π4.

How To

Given an expression of the form f1(f(θ)) where f(θ)=sin(θ), cos(θ), or tan(θ), evaluate.

  1. If θ is in the restricted domain of f, then f1(f(θ))=θ.
  2. If not, then find an angle φ within the restricted domain of f such that f(φ)=f(θ). Then f1(f(θ))=φ.

Example 5: Using Inverse Trigonometric Functions

Evaluate the following:

  1. sin1(sin(π3))
  2. sin1(sin(2π3))
  3. cos1(cos(2π3))
  4. cos1(cos(π3))

Try It #5

Evaluate tan1(tan(π8)) and tan1(tan(11π9)).

Evaluating Compositions of the Form f1(g(x)) (Optional)

Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form f1(g(x)). For special values of x, we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is θ, making the other π2θ. Consider the sine and cosine of each angle of the right triangle in Figure 9.

An illustration of a right triangle with angles theta and pi/2 - theta. Opposite the angle theta and adjacent the angle pi/2-theta is the side a. Adjacent the angle theta and opposite the angle pi/2 - theta is the side b. The hypoteneuse is labeled c.

Figure 9: Right triangle illustrating the cofunction relationships

Because cos(θ)=bc=sin(π2θ), we have sin1(cos(θ))=π2θ if 0θπ. If θ is not in this domain, then we need to find another angle that has the same cosine as θ and does belong to the restricted domain; we then subtract this angle from π2. Similarly, sin(θ)=ac=cos(π2θ), so cos1(sin(θ))=π2θ if π2θπ2. These are just the function-cofunction relationships presented in another way.

How To

Given functions of the form sin1(cos(x)) and cos1(sin(x)), evaluate them.

  1. If x is in [0,π], then sin1(cos(x))=π2x.
  2. If x is not in [0,π] then find another angle y in [0,π] such that cos(y)=cos(x).
     sin1(cos(x))=π2y.
  3. If x is in [π2,π2], then cos1(sin(x))=π2x.
  4. If x is not in [π2,π2], then find another angle y in [π2,π2] such that sin(y)=sin(x).
     cos1(sin(x))=π2y.

Example 6: Evaluating the Composition of an Inverse Sine with a Cosine

Evaluate sin1(cos(13π6))

  1. by direct evaluation.
  2. by the method described previously.

Try It #6

Evaluate cos1(sin(11π4)).

Evaluating Compositions of the Form f(g1(x))

To evaluate compositions of the form f(g1(x)), where f and g are any two of the functions sine, cosine, or tangent and x is any input in the domain of g1, we have exact formulas, such as sin(cos1(x))=1x2. When we need to use them, we can derive these formulas by using the trigonometric relations between the angles and sides of a right triangle, together with the use of the Pythagorean relation between the lengths of the sides. We can use the Pythagorean identity, sin2(x)+cos2(x)=1, to solve for one when given the other. We can also use the inverse trigonometric functions to find compositions involving algebraic expressions.

Example 7: Evaluating the Composition of a Sine with an Inverse Cosine

Find an exact value for sin(cos1(45)).

Try It #7

Evaluate cos(tan1(512)).

Example 8: Evaluating the Composition of a Sine with an Inverse Tangent

Find an exact value for sin(tan1(74)).

Try It #8

Evaluate cos(sin1(79)).

Example 9: Finding the Cosine of the Inverse Sine of an Algebraic Expression

Find a simplified expression for cos(sin1(x3)) for 3x3.

Try It #9

Find a simplified expression for sin(tan1(4x)) for 14x14.

Access this online resource for additional instruction and practice with inverse trigonometric functions.

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Key Concepts

  • An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
  • Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
  • For any trigonometric function f(x), if x=f1(y), then f(x)=y. However, f(x)=y only implies x=f1(y) if x is in the restricted domain of f.
  • Special angles are the outputs of inverse trigonometric functions for special input values; for example, π4=tan1(1) and π6=sin1(12).
  • A calculator will return an angle within the restricted domain of the original trigonometric function.
  • Inverse functions allow us to find an angle when given two sides of a right triangle.
  • In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, sin(cos1(x))=1x2.
  • If the inside function is a trigonometric function, then the only possible combinations are sin1(cos(x))=π2x if 0xπ and cos1(sin(x))=π2x if π2xπ2.
  • When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function.
  • When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides.

Glossary

arccosine
another name for the inverse cosine; arccos(x)=cos1(x)
arcsine
another name for the inverse sine; arcsin(x)=sin1(x)
arctangent
another name for the inverse tangent; arctan(x)=tan1(x)
inverse cosine function
the function cos1(x), which is the inverse of the cosine function and the angle that has a cosine equal to a given number
inverse sine function
the function sin1(x), which is the inverse of the sine function and the angle that has a sine equal to a given number
inverse tangent function
the function tan1(x), which is the inverse of the tangent function and the angle that has a tangent equal to a given number