7. The Inverse Trigonometric Functions

by M. Bourne

In the section Trigonometric Functions of Any Angle, we were solving questions like

"Find 2 angles whose cosine is 0.7."

This question involved the use of the cos^-1 button on our calculators. We found cos^-10.7 and then considered the quadrants where cosine was positive. Remember that the number we get when finding the inverse cosine function, cos^-1, is an angle.

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Now we turn our attention to all the inverse trigonometric functions and their graphs. It is good to have a sense of these graphs so that you know why there are restrictions on the values that we find on our calculators.

The Inverse Cosine Function (arccos)

[I have mentioned elsewhere why it is better to use arccos than cos^-1 when talking about the inverse cosine function.]

Let's first recall the graph of y = cos x (which we met in Graphs y = a cos x) so we can see where the graph of y = arccos x comes from.

cos x

We now choose the portion of this graph from x = 0 to x = π.

The graph of the inverse of cosine x is found by reflecting the graph of cos x through the line y = x.

We now reflect every point on this portion of the cos x curve through the line y = x.

The result is the graph y = arccos x:

arccos x

That's it for the graph - it does not extend beyond what you see here. (If it did, there would be multiple values of y for each value of x and then we would no longer have a function.)

The domain (the possible x-values) of arccos x is

-1 ≤ x ≤ 1

The range (of y-values for the graph) for arccos x is

0 ≤ arccos x ≤ π

The Inverse Sine Function (arcsin)

We define the inverse sine function as

y = arcsin x for

where y is the angle whose sine is x. This means that

x = sin y

The graph of y = arcsin x

Let's see the graph of y = sin x first and then derive the curve of y = arcsin x.

sin x

As we did previously , if we reflect the indicated portion of y = sin x through the line y = x, we obtain the graph of y = arcsin x:

arcsin x

Once again, what you see is what you get. The graph does not extend beyond the indicated boundaries of x and y.

The domain (the possible x-values) of arcsin x is

-1 ≤ x ≤ 1

The range (of y-values for the graph) for arcsin x is

-π/2 ≤ arcsin x ≤ π/2

The Inverse Tangent Function (arctan)

As a reminder, here is the graph of y = tan x, that we met before in Graphs of tan, cot, sec and csc.

tan x

Reflecting this portion of the graph in the line y = x, we obtain the graph of y = arctan x:

arctan x

This time the graph does extend beyond what you see, in both the negative and positive directions of x.

The domain (the possible x-values) of arctan x is

All values of x

The range (of y-values for the graph) for arctan x is

-π/2 ≤ arctan x ≤ π/2

Numerical Examples of arcsin, arccos and arctan

Using a calculator in radian mode, we obtain the following:

arcsin0.6294 = sin ^-10.6294 = 0.6808
arcsin(-0.1568) = sin ^-1(-0.1568) = -0.1574
arccos(-0.8026) = cos ^-1(-0.8026) = 2.5024
arctan(-1.9268) = tan ^-1(-1.9268) = -1.0921

Note that the calculator will give the values that are within the defined range for each function.

The answers in each case are angles.

The Inverse Secant Function (arcsec)

The graph of y = sec x, that we met before in Graphs of tan, cot, sec and csc:

sec x

The graph of y = arcsec x:

arcsec x

The domain of arcsec x is

All values of x, except -1 ≤ x ≤ 1

The range of arcsec x is

0 < arcsec x < π, arcsec x ≠ π/2

The Inverse Cosecant Function (arccsc)

The graph of y = csc x, that we met before in Graphs of tan, cot, sec and csc:

cscx

The graph of y = arccsc x:

arccsc x

The graph extends in the negative and positive x-directions.

The domain of arccsc x is

All values of x, except -1 ≤ x ≤ 1

The range of arccsc x is

-π/2 ≤ arccsc x ≤ π/2, arccsc x ≠ 0

The Inverse Cotangent Function (arccot)

The graph of y = cot x, that we met before in Graphs of tan, cot, sec and csc:

cot x

The graph of y = arccot x:

arccot x

The graph extends in the negative and positive x-directions.

The domain of arccot x is

All values of x

The range of arccot x is

0 < arccot x < π

Numerical Exercises

1. Find cos (sin ^-10.5)

Answer: The value of the portion in brackets is an angle.

Noting the range for inverse sine function, we get

sin ^-10.5 =

Hence cos (sin ^-10.5) = cos () = 0.8660

This final answer is a number, not an angle.

2. Write down the meaning (in words) of y = 2 sin ^-1x

Answer: y is twice the angle whose sine is x.

3. Evaluate sin ^-10

Answer: sin ^-10 = 0

4. Evaluate sec ^-12

Answer: sec ^-12 =

5. Evaluate cos [ tan ^-1(-1) ]

Answer: tan ^-1(-1) = − π /4

cos[-π/4] =

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