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