The arcsine, arccosine and arctangent functions

The arcsine function

Background: The arcsine function is the inverse of the sine function (as long as the sine function is restricted to a certain domain). Click here for a review of inverse functions.

Recall that the sine function takes an angle x as input and returns the sine of that angle as output:

For example if 30° is the input then 0.5 is the output. Here we want to create the inverse function that would take 0.5 as input and return 30° as output. But there is a problem. Notice that there are many angles whose sine is 0.5:

We say that this mapping is many-to-one. This means that the inverse mapping would be one-to-many and therefore would not satisfy the ‘one range value’ requirement for a mapping to be a function. To fix this problem we introduce both a relation called Arcsine (with capital A and abbreviation Arcsin) and a function called arcsine (with lower case ‘a’ and abbreviation arcsin). Here is how they are defined:

Definition: The Arcsine of x, denoted Arcsin(x), is defined as ‘the set of all angles whose sine is x’. It is a one-to-many relation. Here is an example:

Definition: The arcsine of x, denoted arcsin(x), is defined as is defined as ‘the one angle between −π/2 and +π/2 radians (or between −90° and +90°) whose sine is x’. It is a one-to-one function. Here is an example:

Definition: Of all the values returned by the Arcsine relation, the one that is the same as the value returned by the arcsine function is called the principal value of the Arcsine relation. (An example is the value 30° shown above in red.)

Graph: The red curve in the graph to the right is the arcsine function. Notice that for any x between −1 and +1 it returns a single value between −π/2 and +π/2 radians.

If we add the gray curve to the red curve then we get a graph of the Arcsine relation. A vertical line drawn anywhere between x = −1 and +1 would touch this curve at many places and this means that the Arcsine relation would return many values.

In this graph the angle y is measured in radians. If you want to measure y in degrees then simply change the vertical scale so that y runs from −360° to 360° instead of from −2π to 2π radians; the shape of the graph is otherwise unchanged.

If you compare the Arcsine graph to the sine graph then you see that one can be gotten from the other by interchanging the horizontal and vertical axes.

Domain and range: The domain of the arcsine function is from −1 to +1 inclusive and the range is from −π/2 to π/2 radians inclusive (or from −90° to 90°).

The arcsine function can be extended to the complex numbers, in which case the domain is all complex numbers.

Special values of the arcsine function (Click here for more details)

Solving the Equation sin(θ) = c for θ by using arcsine and Arcsine

Suppose that an angle θ is unknown but that its sine is known to be c. Then finding that angle requires solving this equation for θ:

sin (θ) = c

If this is, say, a simple right triangle problem and we know that the angle θ must be somewhere between 0 and 90° then the solution is the single value:

θ = arcsin (c)

On the other hand, if this is a more advanced problem and we need to find all the possible angles whose sine is c then the solution is the entire set of values:

θ = Arcsin (c)

The solutions in these two cases follow directly from the definitions of the arcsine function and Arcsine relation. Note that if c is greater than 1 or less than −1 then there are no real solutions. However there are complex solutions.

Evaluating Arcsin(c)

If c is a number then the entire set of values of Arcsin(c) can be found using the following procedure. Refer to the graph to the right where the dots are the desired values.

The first value (the principal value), denoted θ_PV , is found by evaluating arcsin(c) with a calculator or with the Algebra Coach.
The second value, called θ₂ , is found by using the symmetry of the Arcsine curve. Notice that the two blue arrows in the graph have the same length. This means that θ₂ is just as far below π as θ_PV is above zero. In formula form:
θ₂ = π − θ_PV
(Click here to see the CAST method for finding θ₂ .)
All the other values above and below these two values can be found from these two values by adding or subtracting multiples of 2π. If we use the integer n to count which multiple then the other values can be gotten from this formula:
For example if we let n = −1 then we get values for the two lowest dots in the graph.
If you are using degrees instead of radians then use the following formulas instead of the previous ones:

How to use the arcsine function in the Algebra Coach

Type arcsin(x) into the textbox, where x is the argument. The argument must be enclosed in brackets.
Set the relevant options:
- Set the arcsin, arccos and arctan option. (The return principal value setting returns one value; the don't evaluate setting is useful if you want all the values of the Arc relation - but you will have to calculate them yourself.)
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option.
- Set the p does / does not represent π option. (If you want arcsine to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate the arcsine of a complex number or of a number bigger than 1.
Click the Simplify button.

Algorithm for the arcsine function

Click here to see the algorithm that computers use to evaluate the arcsine function.

The arccosine function

Background: The arccosine function is the inverse of the cosine function (as long as the cosine function is restricted to a certain domain). Click here for a review of inverse functions.

Recall that the cosine function takes an angle x as input and returns the cosine of that angle as output:

For example if 60° is the input then 0.5 is the output. Here we want to create the inverse function that would take 0.5 as input and return 60° as output. But there is a problem. Notice that there are many angles whose cosine is 0.5:

We say that this mapping is many-to-one. This means that the inverse mapping would be one-to-many and therefore would not satisfy the ‘one range value’ requirement for a mapping to be a function. To fix this problem we introduce both a relation called Arccosine (with capital A and abbreviation Arccos) and a function called arccosine (with lower case ‘a’ and abbreviation arccos). Here is how they are defined:

Definition: The Arccosine of x, denoted Arccos(x), is defined as ‘the set of all angles whose cosine is x’. It is a one-to-many relation. Here is an example:

Definition: The arccosine of x, denoted arccos(x), is defined as ‘the one angle between 0 and π radians (or between 0° and 180°) whose cosine is x’. It is a one-to-one function. Here is an example:

Definition: Of all the values returned by the Arccosine relation, the one that is the same as the value returned by the arccosine function is called the principal value of the Arccosine relation. (An example is the value 60° shown above in red.)

Graph: The red curve in the graph to the right is the arccosine function. Notice that for any x between −1 and +1 it returns a single value between 0 and +π radians.

If we add the gray curve to the red curve then we get a graph of the Arccosine relation. A vertical line drawn anywhere between x = −1 and +1 would touch this curve at many places and this means that the Arccosine relation would return many values.

In this graph the angle y is measured in radians. If you want to measure y in degrees then simply change the vertical scale so that y runs from −360° to 360° instead of from −2π to 2π radians; the shape of the graph is otherwise unchanged.

If you compare the Arccosine graph to the cosine graph then you see that one can be gotten from the other by interchanging the horizontal and vertical axes.

Domain and range: The domain of the arccosine function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°).

The arccosine function can be extended to the complex numbers, in which case the domain is all complex numbers.

Special values of the arccosine function (Click here for more details)

Solving the Equation cos(θ) = c for θ by using arccosine and Arccosine

Suppose that an angle θ is unknown but that its cosine is known to be c. Then finding that angle requires solving this equation for θ:

cos (θ) = c

If this is, say, a simple right triangle problem and we know that the angle θ must be somewhere between 0 and 90° then the solution is the single value:

θ = arccos (c)

On the other hand, if this is a more advanced problem and we need to find all the possible angles whose cosine is c then the solution is the entire set of values:

θ = Arccos (c)

The solutions in these two cases follow directly from the definitions of the arccosine function and Arccosine relation. Note that if c is greater than 1 or less than −1 then there are no real solutions. However there are complex solutions.

Evaluating Arccos(c)

If c is a number then the entire set of values of Arccos(c) can be found using the following procedure. Refer to the graph to the right where the dots are the desired values.

The first value (the principal value), denoted θ_PV , is found by evaluating arccos(c) with a calculator or with the Algebra Coach.
The second value, called θ₂ , is found by using the symmetry of the Arccosine curve. Notice that the two blue arrows in the graph have the same length. This means that θ₂ is just as far below 2π as θ_PV is above zero. In formula form:
θ₂ = 2 π − θ_PV
(Click here to see the CAST method for finding θ₂ .)
All the other values above and below these two values can be found from these two values by adding or subtracting multiples of 2π. If we use the integer n to count which multiple then the other values can be gotten from this formula:
For example if we let n = −1 then we get values for the two lowest dots in the graph.
If you are using degrees instead of radians then use the following formulas instead of the previous ones:

How to use the arccosine function in the Algebra Coach

Type arccos(x) into the textbox, where x is the argument. The argument must be enclosed in brackets.
Set the relevant options:
- Set the arcsin, arccos and arctan option. (The return principal value setting returns one value; the don't evaluate setting is useful if you want all the values of the Arc relation - but you will have to calculate them yourself.)
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option.
- Set the p does / does not represent π option. (If you want arccosine to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate the arccosine of a complex number or of a number bigger than 1.
Click the Simplify button.

Algorithm for the arccosine function

Click here to see the algorithm that computers use to evaluate the arccosine function.

The arctangent function

Background: The arctangent function is the inverse of the tangent function (as long as the tangent function is restricted to a certain domain). Click here for a review of inverse functions.

Recall that the tangent function takes an angle x as input and returns the tangent of that angle as output:

For example if 45° is the input then 1.0 is the output. Here we want to create the inverse function that would take 1.0 as input and return 45° as output. But there is a problem. Notice that there are many angles whose tangent is 1.0:

We say that this mapping is many-to-one. This means that the inverse mapping would be one-to-many and therefore would not satisfy the ‘one range value’ requirement for a mapping to be a function. To fix this problem we introduce both a relation called Arctangent (with capital A and abbreviation Arctan) and a function called arctangent (with lower case ‘a’ and abbreviation arctan). Here is how they are defined:

Definition: The Arctangent of x, denoted Arctan(x), is defined as ‘the set of all angles whose tangent is x’. It is a one-to-many relation. Here is an example:

Definition: The arctangent of x, denoted arctan(x), is defined as ‘the one angle between −π/2 and +π/2 radians (or between −90° and +90°) whose tangent is x’. It is a one-to-one function. Here is an example:

Definition: Of all the values returned by the Arctangent relation, the one that is the same as the value returned by the arctangent function is called the principal value of the Arctangent relation. (An example is the value 45°shown above in red.)

Graph: The red curve in the graph to the right is the arctangent function. Notice that for any x it returns a single value between −π/2 and +π/2 radians.

If we add the gray curves to the red curve then we get a graph of the Arctangent relation. A vertical line drawn anywhere would touch this set of curves at many places and this means that the Arctangent relation would return many values.

In this graph the angle y is measured in radians. If you want to measure y in degrees then simply change the vertical scale so that y runs from −180° to 180° instead of from −π to π radians; the shape of the graph is otherwise unchanged.

If you compare the Arctangent graph to the tangent graph then you see that one can be gotten from the other by interchanging the horizontal and vertical axes.

Domain and range: The domain of the arctangent function is all real numbers and the range is from −π/2 to π/2 radians exclusive (or from −90° to 90°).

The arctangent function can be extended to the complex numbers, in which case the domain is all complex numbers.

Special values of the arctangent function (Click here for more details)

Solving the Equation tan(θ) = c for θ by using arctangent and Arctangent

Suppose that an angle θ is unknown but that its tangent is known to be c. Then finding that angle requires solving this equation for θ:

tan (θ) = c

If this is, say, a simple right triangle problem and we know that the angle θ must be somewhere between 0 and 90° then the solution is the single value:

θ = arctan (c)

On the other hand, if this is a more advanced problem and we need to find all the possible angles whose tangent is c then the solution is the entire set of values:

θ = Arctan (c)

The solutions in these two cases follow directly from the definitions of the arctangent function and Arctangent relation.

Evaluating Arctan(c)

If c is a number then the entire set of values of Arctan(c) can be found using the following procedure. Refer to the graph to the right where the dots are the desired values.

The first value (the principal value), denoted θ_PV , is found by evaluating arctan(c) with a calculator or with the Algebra Coach.
All the other values above and below this value can be found by using the fact that adjacent values are separated from each other by a distance of π. If we use the integer n to count multiples of π then the other values can be gotten from this formula:
θ = θ_PV + π n
(Click here to see the CAST method for finding θ₂ .)
If you are using degrees instead of radians then use the following formulas instead of the previous ones:
θ = θ_PV + 180° · n

How to use the arctangent function in the Algebra Coach

Type arctan(x) into the textbox, where x is the argument. The argument must be enclosed in brackets.
Set the relevant options:
- Set the arcsin, arccos and arctan option. (The return principal value setting returns one value; the don't evaluate setting is useful if you want all the values of the Arc relation - but you will have to calculate them yourself.)
- Set the exact / floating point option. (Exact mode lets you use special values.)
- Set the degree / radian mode option.
- Set the p does / does not represent π option. (If you want arctangent to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate the arctangent of a complex number.
Click the Simplify button.

Algorithm for the arctangent function

Click here to see the algorithm that computers use to evaluate the arctangent function.

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