MATH 155 SUPPLEMENTAL NOTES 7

REVIEW OF INVERSE TRIGONOMETRIC FUNCTIONS

THE DOMAIN AND RANGE OF THE SIX BASIC TRIG AND INVERSE TRIG FUNCTIONS

Remember that the six basic trig functions are not one-to-one, so to define the inverse functions for these trig functions, we must restrict their domain.

 

DOMAIN

RANGE

y = sin x

[-p /2, p /2]

[-1,1]

y = sin -1 x

[-1,1]

[-p /2, p /2]

y = cos x

[0, p ]

[-1, 1]

y = cos -1 x

[-1, 1]

[0, p ]

y = tan x

(-p /2, p /2)

(-¥ ,¥ )

y = tan -1 x

(-¥ ,¥ )

(-p /2, p /2)

y = cot x

(0, p )

(-¥ ,¥ )

y = cot -1 x

(-¥ ,¥ )

(0, p )

y = sec x

[0, p /2) È (p /2, p ]

(-¥ , -1] È [1, ¥ )

y = sec -1 x

(-¥ , -1] È [1, ¥ )

[0, p /2) È (p /2, p ]

y = csc x

[-p /2, 0) È (0, p /2]

(-¥ , -1] È [1, ¥ )

y = csc -1 x

(-¥ , -1] È [1, ¥ )

[-p /2, 0) È (0, p /2]

 

Here are the graphs of six basic trig functions and their related inverse trig functions.

y = sin x

y = sin -1 x

y = cos x

y = cos -1 x

y = tan x

y = tan -1 x

y = sec x

y = sec -1 x

y = csc x

y = csc -1 x

y = cot x

y = cot -1 x

USING REFERENCE TRIANGLES TO DETERMINE THE VALUES OF INVERSE TRIG FUNCTIONS

EXAMPLE 1: Determine the value of tan -1(-1).

SOLUTION:

Let us look at the reference triangle for the angle a = tan -1 (-1) or tan a = -1.

What angle, a, has a triangle that has side lengths of 1 and -1, and the length of the hypotenuse is the

The angle, a, is - p /4.

 

 EXAMPLE 2: Determine the value of the following inverse trig function.

SOLUTION:

Let us look at the reference triangle for the angle

What angle a has this triangle as its right triangle?

The angle that produces this triangle is a = p / 3.

 

EXAMPLE 3: Determine the value of the following inverse trig function.

 SOLUTION:

Let us look at the reference triangle for the angle

What angle a has this triangle as its right triangle?

The angle that produces this triangle is a = p / 6.

 

TRIGONOMETRIC FUNCTION VALUES

Sometimes you will be given a value of an angle in terms of an inverse trig function, and you will be asked to determine the values of the other trig functions. To accomplish this task, we must again draw a reference triangle, and then find the required values.

EXAMPLE 4:

Given a = tan -1(3/4), find sin a, cos a, cot a, sec a, csc a.

SOLUTION:

First of all, let us draw a reference triangle for

a = tan -1(3/4) or tan a = 3/4.

Next, we will have to determine the value of c. We will use the Pythagorean Theorem to do this.

Using the definition of the trig functions, we will find the rest of the trig values.

sin a = 3/5

cos a = 4/5

cot a = 4/3

sec a = 5/4

csc a = 5/3

 EXAMPLE 5:

Given a = sin -1(1/2), find cos a, tan a, cot a, sec a, csc a.

SOLUTION:

First of all, let us draw the reference triangle for

a = sin -1(1/2) or sin a = 1/2.

Next, we must determine the length of side a by using the Pythagorean Theorem.

Using the definitions of the trig functions, we will find the rest of the values.

 EXAMPLE 6:

Given a = cos -1 x, find sin a , sec a , csc a , tan a , cot a .

SOLUTION:

Let us draw the reference triangle for the angle

a = cos -1 x or cos a = x.

Now, let us solve for side b.

Finally, let find the values of the remaining trig functions.

You are probably wondering why I wanted do some examples of finding trig values of an angle given in terms of a variable. Later on in techniques of integration, we will learn how to use a method called integration by trig substitution. In this method, one of the last few steps is to construct a reference triangle based on the trig substitution that we used, then use the this triangle to substitute back the original variable. We will talk more about this topic in later supplemental notes. Let us do a couple more examples.

EXAMPLE 7:

Given a = tan -1 (1 - x), find sin a , cos a , cot a , sec a , csc a .

SOLUTION:

Here is the reference triangle for the angle a = tan -1 (1 - x) or tan a = 1- x.

Next, we must determine what c is.

Finally to determine the rest of the trig function's values.

EXAMPLE 8:

SOLUTION:

The first thing we will do to solve this problem is to construct a reference triangle for the angle

Next, we need to determine the value of side b.

Finally, we will find the value of tan a . Remember that a is the value that we defined above. Therefore,

EXAMPLE 9:

SOLUTION:

First of all, we will construct the reference triangle for the angle a .

Next, we need to determine the value of c.

Finally, we will find sin a , where a is what we defined above.

 

Work through all of these examples, taking care to use the definitions of the trig functions. As I have stated before, we will be using this concept in the technique of integration called trig substitution, so it would be good time to get this concept down. If you have an questions or problems, feel free to contact me.

 

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