8-4 Relationships Among the Functions!

Today, I'm going to teach you about the relationships among the functions. This is very confusing, but I'm going to try to teach you about it anyway.

There are only 4 steps in working these problems, but they are very loosley followed. These 4 steps are:

1. The first thing you do is to do all the possible algebra to the problem.
2. Once you do everything possible, you would first try to use your pythagorean identities. After that you would move everything to sin and cos.
3. The next thing you would do is do algebra again.
4. Once you do all of that you keep repeating steps 1 through 3 until your problem is completely simplified.

Before I work any examples, I am going to list the pythagorean identities and how you change things to sin and cos.

Pythagorean Identities:

• sin^2x+cos^2x=1
• 1+tan^2x=sec^2x
• 1+cot^2x=csc^2x

Changing to Sin and Cos

• cscx=1/sinx
• tanx=sinx/cosx
• cotx=cosx/sinx
• secx=1/cosx

Now I am going to work a few examples!

Example 1:cos^2x+sin^2x

1. There is no possible algebra that can be done to this problem.
2. You are first going to look for identities, which you can tell that this problem is one of the pythagorean identities which means it will equal to 1.

Your answer is going to be: 1

Example 2:cotx secx sinx

1. There is no possible algebra that can be done to this problem.
2. There are no possible identities that can be done to this problem. So you would change it to sin and cos. When you refer to the note above you can tell that cotx is going to equal to cosx/sinx, secx is going to equal to 1/cosx, and sinx is going to stay the same.
3. Your problem is then going to look like this cosx/sinx 1/cosx sinx/1. Now when your doing simple algebra you would see that the cosx's are going to cancel out and so are the sinx's. That is going to leave you with 1/1, which then equals to 1.

Your answer is going to be: 1

Example 2:1+tan^2x

1. There is no possible algebra that can be done to this problem.
2. This problem is one of your pythagorean identities, which is going to make your problem equal to secx^2. That is then going to be your answer.

Your answer is going to be: secx^2

And that is how you work these types of problems!

--Halie! :)