the arcsin function | the arccos function | the arctan function |

Recall that the sin function takes an angle

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

In this graph the angle

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.

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

Suppose that an angle

sin (If this is, say, a simple right triangle problem and weθ) =c

On the other hand, if this is a more advanced problem and we need to findθ= arcsin (c)

The solutions in these two cases follow directly from the definitions of the arcsin function and Arcsin relation. Note that ifθ= Arcsin (c)

If

- The first value (the principal value), denoted
*θ*, is found by evaluating arcsin(_{PV}*c*) with a calculator or with the Algebra Coach. - The second value, called
*θ*_{2}, 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*θ*_{2}is just as far below π as*θ*is above zero. In formula form:_{PV}

(Click here to see the CAST method for finding*θ*_{2}= π −*θ*_{PV}*θ*_{2}.) - 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:

- 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.

- Set the
- Click the Simplify button.

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

Recall that the cos function takes an angle

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

In this graph the angle

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.

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

Suppose that an angle

cos (If this is, say, a simple right triangle problem and weθ) =c

On the other hand, if this is a more advanced problem and we need to findθ= arccos (c)

The solutions in these two cases follow directly from the definitions of the arccos function and Arccos relation. Note that ifθ= Arccos (c)

If

- The first value (the principal value), denoted
*θ*, is found by evaluating arccos(_{PV}*c*) with a calculator or with the Algebra Coach. - The second value, called
*θ*_{2}, 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*θ*_{2}is just as far below 2π as*θ*is above zero. In formula form:_{PV}

(Click here to see the CAST method for finding*θ*_{2}= 2 π −*θ*_{PV}*θ*_{2}.) - 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:

- 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.

- Set the
- Click the Simplify button.

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

Recall that the tan function takes an angle

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

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.

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

Suppose that an angle

tan (If this is, say, a simple right triangle problem and weθ) =c

On the other hand, if this is a more advanced problem and we need to findθ= arctan (c)

The solutions in these two cases follow directly from the definitions of the arctan function and Arctan relation.θ= Arctan (c)

If

- The first value (the principal value), denoted
*θ*, is found by evaluating arctan(_{PV}*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:

(Click here to see the CAST method for finding*θ*=*θ*+ π_{PV}*n**θ*_{2}.) - If you are using degrees instead of radians then use the following formulas
instead of the previous ones:
*θ*=*θ*+ 180° ·_{PV}*n*

- 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.

- Set the
- Click the Simplify button.

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