the arcsine function     the arccosine function     the arctangent function  


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:
The principal value of the Arcsin is the value shown in red
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.



How to use the arcsine function in the Algebra Coach


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:
The principal value of the Arccos is the value shown in red
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.



How to use the arccosine function in the Algebra Coach


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:
The principal value of the Arctan is the value shown in red
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.



How to use the arctangent function in the Algebra Coach


Algorithm for the arctangent function

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





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