Relationships Among The Functions

Yep.

This week, we learned how to simplify trig equations. It can be extremely useful, because icky problems like cos^3(y)+cos(y)sin^2(y) can be simplified to just cos(y).

Now the useful things we use to do this:

When a sin^2Θ and a cos^2Θ love each other very much... and are magically added together with a plus sign, they apparent equal 1.

That (loosely) is a Pythagorean Property. The Pythagorean Properties are:

sin^2Θ + cos^2Θ = 1

1 + tan^2Θ = sec^2Θ

1 + cot^2Θ = csc^2Θ

Other things you should know:

cscΘ = 1/sinΘ

secΘ = 1/cosΘ

cotΘ = cosΘ/sinΘ

tanΘ = sinΘ/cosΘ

I usually remember cot and tan this way: since cot starts with a c, you write cos first over sin, and tan doesn't start with a c, so sin goes on top.

Even more things you should know:
sinΘ = cos(90°-Θ) and cosΘ = sin(90°-Θ)

tanΘ = cot(90°-Θ) and cotΘ = tan(90°-Θ)

secΘ = csc(90°-Θ) and cscΘ = sec(90°-Θ)

It's not really "and" but I just put it that way because it's how it's easier for me to remember, it's just flipped.

Steps to loosely follow:

1) You should try any algebra that you can do (factoring, cancelling, dividing, multiplying, etc.)

2) Check your identities - If the Pythagorean Properties don't work, convert everything to sin and cos

3) Try algebra again

4) Rinse and repeat. Continue until you can't do anything else and it's completely simplified.

Ex.

cos^3(y)+cos(y)sin^2(y)

Algebra- You can factor out a cos(y):

cos(y) [cos^2(y) + sin^2(y)]

Identities- The pythagorean property sin^2Θ + cos^2Θ = 1:

cos(y) (1)

Algebra - Multiply it out

= cos(y)

Ex.

tan(x)+cot(x)/sec^2(x)

Algebra - There's nothing you can really do here with algebra

Identities- You can convert everything to sin and cos because the Pythagorean Properties don't help here:

cos^2(x) [sin(x)/cos(x)+cos(x)/sin(x)]

Algebra- To add, you have to get a common denominator:

cos^2(x) [sin^2(x)/cos(x)sin(x) + cos^2(x)/cos(x)sin(x)]

Algebra- Adding together:

cos^2(x) [sin^2(x) + cos^2(x)/cos(x)sin(x)]

Identities- Pythagorean Identity sin^2(x) + cos^2(x) = 1:

cos^2(x) [1/cos(x)sin(x)], or cos^2(x)/cos(x)sin(x)

Algebra- cos(x) cancels so you're left with:

cos(x)/sin(x), which is equal to cot(x)

So tan(x)+cot(x)/sec^2(x) = cot(x)