the arcsine function | the arccosine function | the arctangent function |

Recall that the sine function takes an angle

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

In this graph the angle

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.

The arcsine 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 arcsine function and Arcsine 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 Arcsine 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 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.

- Set the
- Click the Simplify button.

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

Recall that the cosine function takes an angle

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

In this graph the angle

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.

The arccosine 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 arccosine function and Arccosine 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 Arccosine 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 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.

- Set the
- Click the Simplify button.

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

Recall that the tangent function takes an angle

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

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.

The arctangent 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 arctangent function and Arctangent 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 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.

- Set the
- Click the Simplify button.

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

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