Relationships among the functions

Soo today we will be learning about relationships among the different functions such as cos tan and sin. Next i will explain in deep details how you will deal with these functions.

-The first step is to use your algebra skills to add,subtract,multiply,or divide any of the fucntions that can be simplified.

-The next step is to use your identities to simplify.

-Next you will do algebra again to try to simplify more.

-You will repeat the same steps over and over again till your equation is in simplest form.

PYTHAGOREAN

sin^2x+cos^2x=1

1+cot^2x=csc^2x

1+tan^2x=sec^2x

Recipricles

cscx=1/sinx

tanx=sinx/cosx

secx=1/cosx

cotx=cosx/sinx

So now that i have explained to you how to solve/simplify these function and showed you the identities I will show you how to work them.

Examples:

(sec^2x-1)(csc^2x-1)

(tan^2x)(cot^2x)

(sin^2x/cos^2x)(cos^2x/sin^2x)

0

1+cot^2a reduces to 1/sin^2a

8-4

So, today we are going to learn about relationships in trig functions. This is one of those things that will get easier and easier as you do it. There are steps to these kinds of problems although you follow them very loosely. They are:

1. Do as much basic algebra as you can.

2. Identity: try to use one of your Pythagorean identities if you can, if not, change everything to sine and cosine.

3. Do algebra again.

4. Follow all the steps again.

Some things that you will need to know are:

Pythagorean Identities:

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

Formulas for changing to Sine and Cosine:

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

Now for the example:

Sec^2x-1

Step 1: there is no possible algebra that can be done to this problem

Step 2: You can use the pythagorean identity "1+tan^2x=sec^2x". Solve for Sec^2x- 1 which would give you tan^2x.

Your answer would be tan^2x!!!

I hope you learned something! OH, and i know it's depressing that this isn't colorful, but I'm super tired, so next time guys :)))

Carlaaaay!

8-4 Relationships Among Functions

This week I am going to teach you how to solve problems by using relationships among trig functions. This section can look extremely intimidating. Do not worry about it. After you do a few problems, you will get the hang of it. There are a multitude of formulas that you can use to help you evaluate these problems. Here is a list of the formulas:
Reciprocal Relationship Formulas:
csc(theta)=1/sin(theta) sec(theta)=1/cos(theta) cot(theta)=1/tan(theta)
tan(theta)=sin(theta)/cos(theta) cot(theta)=cos(theta)/sin(theta)
Pythagorean Relationship Formuals:
sin^2(theta)+cos^2(theta)=1 1+tan^2(theta)=sec^2(theta) 1+cot^2(theta)=csc^2(theta)
Cofunction Relationship Formulas:
sin(theta)=cos(90 degrees-theta) and cos(theta)=sin (90 degrees-theta)
tan(theta)=cot(90 degrees-theta) and cot(theta)=tan (90 degrees-theta)
sec(theta)=csc(90 degrees-theta) and csc(theta)=sec (90 degrees-theta)

The steps are as follows:
(Follow these loosely)
1. Algebra
2. Identities(try Pythagorean first, then change everything to sin and cos if it helps)
3. Algebra
4. Continue with steps 1-3
Examples:
1. (1-cosx)(1+cosx)
• 1+cosx-cosx-cos^2(x)
• 1-cos^2(x)
• sin^2(x)
2. (1-cosx)(1+secx)cosx
• (1-cosx)(1+1/cosx)cosx
• 1+(1/cosx)-cos-(cos/cos)
• 1+(1/cosx)-cosx-1
• (1/cosx)-cosx
• (1/cosx)-(cos^2/cos)
• (1-cos^2/cos)
• (sin^2/cos)
-Braxton

8-4 Relationships Among the Functions

I am going to simplify Tan^2x - Sec^2x; yes this is a relatively simple example on how to do this but it hits the basics.

First, recognize that there is nothing algebraically that can be done to those functions, so move on to step two.

1+Tan^2=Sec^2 is a Pythagorean Identity, so what you would do is make 1+Tan^2=Sec^2 look like Tan^2-Sec^2 and that is done using some algebra.

Once that is done you should have Tan^2x-Sec^2x = -1; have simplified the problem.

~Parrish Masters

8-4

Relationships Among the Functions

When solving the problems in section 8-4, you must you several formulas that help you evaluate relationships among all the functions. The following are the formulas you can use:

1) sin x/cos x=tan x 2) cos x/sin x=cot x

Reciprocal Relationship Formulas:
3) csc(theta)=1/sin(theta) 4) sec(theta)=1/cos(theta) 5) cot(theta)=1/tan(theta)

Pythagorean Relationship Formuals:
6) sin^2(theta)+cos^2(theta)=1 7) 1+tan^2(theta)=sec^2(theta) 8) 1+cot^2(theta)=csc^2(theta)

*Note that you don't change anything with relationships when you're dealing with negatives.

The following are steps to follow when solving these problems:
1) Algebra
2) Identities: Try Pythagorean theorem, then move everything to sin and cos if that will help.
3) Algebra
4) Continue with steps 1 through 3

Here's some examples(:

1) (sec x-1) (sec x+1)
-(sin x/cos x tan x)/(sin x/cos x(sin x/cos x))
-sin x/sin x
=1

2) Verify tan x sin x+cos x=sec x.
-sin x/cos x(sin x)+cos x
-1/1(sin^2x/cos x)+(cos x/1)cos x/cos x
-(sin^2x/cos x)+(cos^2x/cos x)=(sin^2+cos^2x/cos x)
-1/cos x
=sec x

---Jordan Duhon

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! :)

8-4

8-4 is on the relationships among trig functions.
There are several formulas you need to know for this section:

• Cscx=1/sinx
• Secx=1/cosx
• Cotx=cosx/sinx
• Tanx=sinx/cosx
• sin^2x+cos^2x=1
• 1+tan^2x=sec^2x
• 1+cot^2x=csc^2x
• sinx=cos(90 degrees - theta)
• tanx=cot(90 degrees - theta)
• secx=csc(90 degrees - theta)
• cosx=sin(90 degrees - theta)
• cotx=tan(90 degrees - theta)
• cscx=sec(90 degrees - theta)

Steps:
1. Algebra
2. Identities - try pythagorean theorem then move everything to sin and cos.
3. Algebra
4. Continue with steps 1-3

Ex. 1 sin^2x(1+cot^2x)
1. Algebra: You can't do any algebra with this problem.
2. Identities: You can use one of your pythagorean theorems. So you end up with sin^2x(csc^2x).
3. Algebra: You still can't do any algebra with this problem.
4. Steps 1-3: Now you can change csc^2x to 1/sin^2x. That gives you sin^2x(1/sin^2x). Now you can cancel out the sin^2x leaving you with 1. So your answer is 1.

Ex. 2 sinx/cosxtanx
1. Algebra: You can't do any algebra with this problem.
2. Identities: You can change tanx to sinx/cosx. So you get sinx/cosx(sinx/cosx).
3. Algebra: You can cancel out the cosx leaving you with sinx/sinx.
4. Steps 1-3: Your final answer is 1.

Amber :)