8-4

8-4 Relationships Among the Functions

Today I'm doing my blog on 8-4 because I can guarantee you that I bombed this stuff on my test, and I need all the practice I can get. The following are all the relationship equations you will use in solving the problems in this section.

-sin x/cos x= tan x cos x/sin x= cot x

-Reciprocal Relationships:
csc(theta)=1/sin(theta) sec(theta)=1/cos(theta) cot(theta)=1/tan(theta)

-Pythagorean Relationships:
sin^2(theta)+cos^2(theta)=1 1+tan^2(theta)=sec^2(theta) 1+cot^2(theta)=csc^2(theta)

-Cofunction Relationships:
sin(theta)=cos(90-theta) & cos(theta)=sin(90-theta)
tan(theta)=csc(90-theta) & cot(theta)=tan(90-theta)
sec(theta)=csc(90-theta) & csc(theta)=sec(90-theta)

Example 1:
sec x - sin x tan x
=1/cos x - sin x (sin x/cos x)
=1-sin^2 x/cos x
=cos^2 x/cos x
=cos x
answer= cos x

Example 2:
Prove cot A(1+tan^2 A)/tan A= csc^2 A
=(cot A) (sec^2 A)/tan A
=(cos^2 A/sin^2 A) (1/cos^2 A)
=1/sin^2 A
=csc^2 A
answer= csc^2 A

*Note that when you're dealing with negatives in these problems, treat them just as they'd be positive, using all of your formulas.

-Jordan Duhon