The arcsin, arccos and arctan functions

The arcsin function

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

Recall that the sin function takes an angle x as input and returns the sin 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 sin 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 define both a relation called Arcsin (with capital A) and a function called arcsin (with lower case a).

Definition: Arcsin(x) is defined as “the set of all angles whose sin is x”. It is a one-to-many relation. Here is an example:

Definition: arcsin(x) is defined as “the one angle between −π/2 and +π/2 radians (or between −90° and +90°) whose sin is x”. It is a one-to-one function. Here is an example:

Definition: Of all the values returned by the Arcsin relation, the one that is the same as the value returned by the arcsin function is called the principal value of the Arcsin relation. (An example is the value shown in red two pictures back.)

Graph: The red curve in the graph to the right is the arcsin 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 Arcsin relation. A vertical line drawn anywhere between x = −1 and +1 would touch this curve at many places and this means that the Arcsin 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 Arcsin graph to the sin 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 arcsin 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 arcsin function can be extended to the complex numbers, in which case the domain is all complex numbers.

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

Solving the Equation sin(θ) = c for θ by using arcsin and Arcsin

Suppose that an angle θ is unknown but that its sin 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 sin 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 arcsin function and Arcsin 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 Arcsin 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 arcsin 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 arcsin to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate the arcsin of a complex number or of a number bigger than 1.
Click the Simplify button.

Algorithm for the arcsin function

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

The arccos function

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

Recall that the cos function takes an angle x as input and returns the cos 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 cos 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 define both a relation called Arccos (with capital A) and a function called arccos (with lower case a).

Definition: Arccos(x) is defined as “the set of all angles whose cos is x”. It is a one-to-many relation. Here is an example:

Definition: arccos(x) is defined as “the one angle between 0 and π radians (or between 0° and 180°) whose cos is x”. It is a one-to-one function. Here is an example:

Definition: Of all the values returned by the Arccos relation, the one that is the same as the value returned by the arccos function is called the principal value of the Arccos relation. (An example is the value shown in red two pictures back.)

Graph: The red curve in the graph to the right is the arccos 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 Arccos relation. A vertical line drawn anywhere between x = −1 and +1 would touch this curve at many places and this means that the Arccos 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 Arccos graph to the cos 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 arccos function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°).

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

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

Solving the Equation cos(θ) = c for θ by using arccos and Arccos

Suppose that an angle θ is unknown but that its cos 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 cos 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 arccos function and Arccos 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 Arccos 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 arccos 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 arccos to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate the arccos of a complex number or of a number bigger than 1.
Click the Simplify button.

Algorithm for the arccos function

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

The arctan function

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

Recall that the tan function takes an angle x as input and returns the tan 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 tan 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 define both a relation called Arctan (with capital A) and a function called arctan (with lower case a).

Definition: Arctan(x) is defined as “the set of all angles whose tan is x”. It is a one-to-many relation. Here is an example:

Definition: arctan(x) is defined as “the one angle between −π/2 and +π/2 radians (or between −90° and +90°) whose tan is x”. It is a one-to-one function. Here is an example:

Definition: Of all the values returned by the Arctan relation, the one that is the same as the value returned by the arctan function is called the principal value of the Arctan relation. (An example is the value shown in red two pictures back.)

Graph: The red curve in the graph to the right is the arctan 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 Arctan relation. A vertical line drawn anywhere would touch this set of curves at many places and this means that the Arctan 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 Arctan graph to the tan 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 arctan function is all real numbers and the range is from −π/2 to π/2 radians inclusive (or from −90° to 90°).

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

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

Solving the Equation tan(θ) = c for θ by using arctan and Arctan

Suppose that an angle θ is unknown but that its tan 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 tan 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 arctan function and Arctan 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 arctan 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 arctan to return special values in radian mode then turn this on.)
- Turn on complex numbers if you want to be able to evaluate the arctan of a complex number.
Click the Simplify button.

Algorithm for the arctan function

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