# Trigonometry : Arcsin, Arccos, Arctan

## Example Questions

### Example Question #1 : Understanding Arcsine, Arccosine, And Arctangent

What is  if  and ?

Explanation:

In order to find  we need to utilize the given information in the problem.  We are given the opposite and adjacent sides.  We can then, by definition, find the  of  and its measure in degrees by utilizing the  function.

Now to find the measure of the angle using the  function.

If you calculated the angle's measure to be  then your calculator was set to radians and needs to be set on degrees.

### Example Question #2 : Arcsin, Arccos, Arctan

For the above triangle, what is  if  and ?

Explanation:

We need to use a trigonometric function to find . We are given the opposite and adjacent sides, so we can use the  and  functions.

### Example Question #3 : Arcsin, Arccos, Arctan

For the above triangle, what is  if  and ?

Explanation:

We need to use a trigonometric function to find . We are given the opposite and hypotenuse sides, so we can use the  and  functions.

### Example Question #1 : Arcsin, Arccos, Arctan

Which of the following is the degree equivalent of the inverse trigonometric function

?

Explanation:

The  is the reversal of the cosine function. That means that if , then .

Therefore,

### Example Question #5 : Arcsin, Arccos, Arctan

Assuming the angle in degrees, determine the value of .

Explanation:

To evaluate , it is necessary to know the existing domain and range for these inverse functions.

Inverse sine:

Inverse cosine:

Inverse tangent:

Evaluate each term.  The final answers must return an angle.

### Example Question #1 : Arcsin, Arccos, Arctan

If

,

what value(s) does  take?

Assume that

No real solution.

Explanation:

If , then we can apply the cosine inverse to both sides:

Since cosine and cosine inverse undo each other; we can then apply sine and secant inverse functions to obtain the solution.

and

and

are the two solutions.

### Example Question #2 : Arcsin, Arccos, Arctan

Calculate .

and

Explanation:

The arcsecant function takes a trigonometric ratio on the unit circle as its input and results in an angle measure as its output. The given function can therefore be rewritten as

and is the angle measure  which, when applied to the cosine function , results in . Notice that the arcsecant function as expressed in the statement of the problem is capitalized; hence, we are looking for the "principal" angle measure, or the one which lies between  and . Since , and since  lies between  and ,

.

### Example Question #1 : Arcsin, Arccos, Arctan

Calculate .

Explanation:

The domain on the argument  for  is

.

The range of the function  is not defined at  or , and so the domain of its inverse, , does not include those values. Hence, we must find the angle  between  and  for which .

Since , the equation  can be rewritten as

,

or

for some x between  and .

Now,  when , since .

Therefore,

.