8-4
Relationships between the Functions
All of the
following
relationships need
to be memorized.
These are the most commonly used functions that appear in many upper
level
calculus courses. Please make every attempt to really
know them!!
Reciprocal
Relationships
| csc q = 1/sin q |
sec q
= 1/cos q |
cot q = 1/tan q |
Negative
Relationships
| sin(-q)
= -sin q |
cos(-q)
= cos q |
| csc (-q) = -csc
q |
sec(- q) = sec
q |
| tan(-q ) = -tan
q |
cot(- q) =
-cot q |
Pythagorean
Relationships
| sin2
q + cos2
q = 1 |
1 + tan2
q = sec2
q |
1 + cot2
q = csc2
q |
Cofunction
Relationships
| sin q = cos(90o
- q ) |
cos q = sin(90o
- q) |
| tan q = cot(90o
- q) |
cot q = tan(90o
- q) |
| sec q = csc(90o
- q) |
csc q = sec(90o
- q) |
Any relationship
that
is true for all values of the variable for which each
side is defined is called an identity.
Each of the relationships above represent a trigonometric
identity. We can use trig identities
to simplify trig expressions, prove other identities and solve more
complex
trig equations.
Simplifying
trig expressions
1) simplify: sin y cot
y
= sin y (cos y/sin y) (Basic
identity)
= cos y (Basic algebra)
2) simplify: csc2
y(1 - cos2 y)
= csc2
y (sin2
y) (pythagorean)
= csc2
y(1/csc2 y)
(reciprocal)
= 1 (basic algebra)
3) simplify: tan x(cot x + tan x)
= 1 + tan2
x (distribute) (tan and cot are reciprocals)
= sec2
x (pythagorean)
4) simplify: cos x cot x + sin x
5) simplify: 
6) simplify: 
Proving
Identities
You
can use trig identities to prove other statements are identities.
You may simplify either the left side of the equation or the right side
of the equal sign. Your goal is to make it look like the other
side
of the equation. You may even work on both sides. It
is probably a good idea to simplify the more complicated side.
Also, remember your basic algebra skills, like getting a common
denominator
and factoring. Keep in mind where you are headed. If cosine
is on the right side, then when working on the left side, try to write
it in terms of cosine. Use the identities. They will
help!
If all else fails, try writing everything in terms of the sine and
cosine.
Good luck!!
1) Prove: tan
x + cot x = tan x csc2
x
Solution:
2) Prove: 
Solution:
3) Prove: 
Solution:
4) Prove: 
Solution:
5) Prove: 
Solution:
That
should give you some techniques in working with identities. Now,
let's head onto more difficult trig equations.
