Inverse Trigonometric Functions Definitions and Examples

Introduction

Learning the inverse trigonometric functions can be daunting, but it’s not as hard as it seems! In this blog post, we’ll go over the definition of inverse trigonometric functions and explore some examples. By the end of this post, you’ll be a pro at inverse trigonometric functions!

What are Inverse Trigonometric Functions?

Inverse trigonometric functions are the inverse of the trigonometric functions. Trigonometric functions are defined as a function that takes an angle and returns a number. The most common trigonometric functions are sine, cosine, and tangent. Inverse trigonometric functions are the inverse of these functions and thus take a number and return an angle.

The most common inverse trigonometric functions are arcsin, arccos, and arctan. These functions are usually abbreviated as sin-1, cos-1, and tan-1, respectively. For example, if we want to find the angle whose cosine is 0.5, we would use the equation cos-1(0.5) = 60°.

Inverse trigonometric functions can be useful in solving problems involving angles and sides of triangles. For example, consider the triangle shown below:

We can see that the length of side b is equal to 2sin(60°). To find the value of x, we can use the equation sin-1(2/b) = x. Thus, sin-1(2/2) = sin-1(1) = 45° and so x = 45°.

Formulas

In mathematics, an inverse trigonometric function is one of the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are usually denoted by sin?1(x), cos?1(x), tan?1(x), cot?1(x), sec?1(x), and csc?1(x).

There are several formulas involving inverse trigonometric functions. For example, the following formula expresses the cosine in terms of the sine:

cos ? ( x ) = sin ? ( ? / 2 ? x ) {displaystyle cos(x)=sin(pi /2-x)}

Similarly, there is a formula for the tangent in terms of the cotangent:

tan ? ( x ) = 1 cot ? ( x ) {displaystyle tan(x)={frac {1}{cot(x)}}}

And there is a formula relating the sine and cosine:

sin ? ( x ) = ± cos ? ( ? / 2 ? x ) {displaystyle sin(x)=pm cos(pi /2-x)}

In addition to these basic formulas, there are

The Three Main Inverse Trigonometric Functions

The three main inverse trigonometric functions are arcsin (sin-1), arccos (cos-1), and arctan (tan-1). These functions “undo” the regular trigonometric functions, allowing us to solve problems involving angles that we could not otherwise solve.

Here are the definitions of the three inverse trigonometric functions:

Arcsin(x) = the angle in radians whose sin is x
Arccos(x) = the angle in radians whose cos is x
Arctan(x) = the angle in radians whose tan is x

Note that you can only find angles using these inverse trigonometric functions if you know the value of sin, cos, or tan for that angle. For example, if someone tells you that sin?=0.5, then you can use arcsin to find ?: arcsin(0.5)=30°. However, if someone just tells you ?=30°, then you cannot use arcsin to find sin? because there is more than one value of sin? that would work (sin 30°=0.5 and sin 210°=-0.5 both work). In this case, you would need to use a different method to find sin?.

Here are some examples of how to use the three inverse trigonometric functions:

Example 1: Find ? given that tan

Arcsine Function

Arcsine Function

The arcsine function is the inverse function of the sine function. It is defined as the inverse of the sine function, which means that it “undoes” the sine function. The domain of the arcsine function is [-1, 1], and its range is [-?/2, ?/2].

To find the arc sine of an angle, we need to use a calculator or a table of values. For example, to find the arc sine of 30°, we would calculate sin-1(0.5). This would give us an answer of 0.523598776 (to 4 decimal places).

The arc sine function can be used to solve problems involving triangles. For example, if we know that one angle in a triangle is 30°, and we want to find the other two angles, we can use the arc sine function. We would set up the equation as follows: sin ? = sin30°. We would then solve for ?, which would give us an answer of 60° and 90°.

Arccosine Function

The arccosine function is defined as the inverse of the cosine function. So, just as the cosine function takes an angle and outputs a number between -1 and 1, the arccosine function takes a number between -1 and 1 and outputs the angle that corresponds to that number.

To visualize this, imagine a circle with a radius of 1 centered at the origin of a coordinate plane. This is called the unit circle. The cosine function outputs a number between -1 and 1 corresponding to the x-coordinate of any point on the unit circle. So, if we input 0 into the cosine function, it will output 1, since that’s the x-coordinate of the point (0,1) on the unit circle. Similarly, if we input ?/2 into the cosine function, it will output 0, since that’s the x-coordinate of the point (0,-1) on the unit circle.

The arccosine function is simply the inverse of this: it takes a number between -1 and 1 and outputs an angle between 0 and ? corresponding to that number. So, if we input 0 into the arccosine function, it will output ?/2, since that’s the angle whose cosine is 0. And if we input 1 into the arccosine function, it will output 0.

Arctangent Function

The arctangent function is the inverse of the tangent function. It returns the angle, in radians, whose tangent is a given number. In other words, it undo what the tangent function does.

The domain of the arctangent function is all real numbers, and its range is (-?, ?).

The graph of the arctangent function looks like this:

As you can see, it is a horizontal line at y = 0 with asymptotes at y = ±?.

To find the arctangent of a number, we use the following formula:

arctan(x) = y

where x is the input and y is the output.

For example, to find the arctangent of 1, we would plug 1 into x and solve for y:

arctan(1) = y
y = tan-1 (1)
y = 45°

Similarly, we can find that the arctangent of -1 is -45°:

arctangent(-1)=-45°

Arccotangent (Arccot) Function

The arccotangent function is the inverse of the tangent function. It is used to find the angle whose tangent is a given number.

The domain of the arccotangent function is all real numbers. The range of the arccotangent function is all real numbers except for those in the interval (-pi/2, pi/2).

To find the arccotangent of a number, we use the following formula:

arccot(x) = y

where y is an angle in radians and x is the tangent of that angle.

For example, to find the arccotangent of 0.5, we would use the following equation:

arccot(0.5) = y

Since we are looking for an angle, we can take the inverse tangent of both sides to solve for y:

y = tan-1(0.5)

= 26.565 degrees

Arcsecant Function

The arcsecant function is the inverse of the secant function. It takes the reciprocal of the secant function and returns the angle in radians.

For example, if we take the arcsecant of 1.5, we are taking the reciprocal of 1.5 and finding the angle in radians that has a secant value of 1.5. From this, we can see that the answer is approximately 0.9553 radians, or 54.7 degrees.

It is important to note that the domain of the arcsecant function is all real numbers except for 1 and -1, as these are the only values for which the secant function does not have a unique inverse.

The range of the arcsecant function is also all real numbers except for 1 and -1, as these are the only values for which the secant function produces undefined results.

Arccosecant Function

The arccosecant function is the inverse of the cosecant function. It is defined as the inverse cosine function, which is used to find the angle that corresponds to a given ratio of the sides of a right triangle.

The arccosecant function can be written in terms of the inverse sine function as follows:

arccosecant(x) = arcsin(1/x)

For example, if we want to find the angle that has a cosecant value of 2, we can use the arccosecant function to find it. We would first calculate 1/2, which is 0.5, and then take the arcsin of that value to find the angle.

In general, the arccosecant function is used when we know the ratio of one side of a right triangle to another, but we do not know the angle that corresponds to that ratio.

Inverse Trigonometric Functions Derivatives

Inverse trigonometric functions are the inverse of the trigonometric functions. Just as with any inverse function, they “undo” the original function. In other words, if you take the derivative of an inverse trigonometric function, you will get back the original function.

The three most common inverse trigonometric functions are arccosine (arccos), arcsine (arcsin), and arctangent (arctan). These functions are sometimes also called cosine inverse (cos-1), sine inverse (sin-1), and tangent inverse (tan-1).

To find the derivative of an inverse trigonometric function, we use the following formula:

Derivative of arccos(x) = – 1 / ?(1 – x2)
Derivative of arcsin(x) = 1 / ?(1 – x2)
Derivative of arctan(x) = 1 / (1 + x2)

Let’s take a look at each of these derivatives in turn. First, we’ll start with the derivative of arccos(x). We can see from the formula that the derivative will be a negative number when x is between -1 and 1. This makes sense because arccos(x) is a decreasing function in this interval.

How to Use Inverse Trigonometric Functions

When working with inverse trigonometric functions, it is important to first identify the function you are working with. The most common inverse trigonometric functions are arcsin, arccos, and arctan. To use these functions, you must first identify the angle you are working with. For example, if you are given an angle of 30 degrees, you would use the function arcsin(30).

Once you have identified the function and angle you are working with, you can begin to solve for the unknown variable. Inverse trigonometric functions allow you to solve for missing sides or angles in a triangle. For example, if you know two sides of a right triangle, but not the angle between them, you can use the arccos function to solve for that angle.

In general, inverse trigonometric functions are relatively easy to use once you understand what they are and how they work. However, it is important to note that these functions only work with right triangles. If you are given an angle or side length that is not part of a right triangle, the inverse trigonometric function will not give you a correct answer.

Domain and Range Of Inverse Trigonometric Functions

Domain and range of inverse trigonometric functions, as the name suggests, are the set of values for which the function produces a result, and the output values of the function respectively. The domain is usually denoted by D, while the range is denoted by R.

The domain of an inverse trigonometric function is the set of all real numbers for which the function produces a result. For example, the domain of cos-1(x) is all real numbers such that -1?x?1. The range of an inverse trigonometric function is the set of all real numbers that the function can produce as a result. For example, the range of cos-1(x) is all real numbers such that 0?y??.

The inverse trigonometric functions are one-to-one functions. This means that each input value has a unique output value, and vice versa. As a result, inverse trigonometric functions have inverses that are also one-to-one functions. The inverse of an inverse trigonometric function is just the original function. For example, (cos-1)?¹(x) = cos(x), since cos(cos-1(x)) = x for all x in the domain of cos-1(x).

History of Inverse Trigonometric Functions

The inverse trigonometric functions go back to the ancient Greek mathematicians. The first recorded use of these functions was by Hipparchus (190-120 BCE), who used them to solve problems in astronomy.

The inverse trigonometric functions were formally defined in the 17th century by Isaac Newton and Gottfried Leibniz. Newton used them to solve problems in calculus, while Leibniz used them to solve problems in differential equations.

These functions have been extensively studied since then and have found many applications in mathematics and physics.

Conclusion

In this article, we looked at the inverse trigonometric functions and how they are defined. We also saw some examples of how to use them. Inverse trigonometric functions can be very useful in solving problems, so it’s worth taking the time to learn about them.