Relationships Among the Functions

This week we learned how to simplify equations with the 6 trig functions in them.

You will need to know theses identities:
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = cos(x) / sin(x)
tan(x) = sin(x) / cos(x)

And then theres the Pythagorean identities:
sin^2(x) + cos^2(x) = 1
1 + tan^2(x) = sec^2(x)
1 + cot^2(x) = csc^2(x)

Steps to try to follow if you need to simplify an equation with multiple trig functions:

1. You try to do anything algebraicly that might help (FOIL, distribute, factor, add, subtract, multiply, divide, etc.)


2. Identify- try to use the pythagorean identities, then convert everything into sine and cosine if it helps.


3. More algebra...


4. Do it again over and over and over until you cannot do it again.

Example 1: (1 -sin(x)) (1 + csc(x)) sin(x)

Foil it : (1 -sin(x) + csc(x) -sin(x)csc(x)) sin(x)

There are no pythagorean identities so switch to sine and cosine: (1 -sin(x) + 1/sin(x) -(sin(x)(1/sin(x)) sin(x)

Divide: (1 -sin(x) + 1/sin(x) - 1) sin(x)

Distribute: sin(x) -sin^2(x) + sin(x)/sin(x) -sin(x)

Add and divide like terms: 1 -sin^2(x)

Pythagorean identity cos^2 = 1 -sin^2: cos^2(x)