5-3: Other Composite Argument Properties

yes...it's almost Thanksgiving break!
This chapter is basically a bunch of formulas.

The Odd-Even Properties

Cosine and Secant are even functions.
cos(-x) = cos x
sec(-x) = sec x

Sine, Cosecant, Tangent, and Cotangent are odd functions.
sin(-x) = -sin x
csc(-x) = -csc x
tan(-x) = -tan x
cot(-x) = -cot x

You could also write sin(-x) = -sin x as sin(x) = -sin(-x). This goes for the other odd functions as well.

It is easier to see why these properties are true by looking at the graphs of some of the functions.

For cos(θ), you can see from the middle graph that if you replace θ with its opposite, you get the same y-value. The cosine function is symmetrical over the y-axis and therefore is an even function.

For sin(θ) and tan(θ), replacing θ with its opposite yields the opposite y value. These are odd functions and are symmetrical about the origin.

Cofunction Properties for Trigonometric Functions

Degrees:
cos θ = sin(90º-θ) and sin θ = cos(90º-θ)

cot θ = tan(90º-θ) and tan θ = cot(90º-θ)

csc θ = sec(90º-θ) and sec θ = csc(90º-θ)

The cofunction properties involve angles and their complementary angles, or angles that add up to 90º.

Radians:
cos x = sin[(π/2)-x)] and sin x = cos[(π/2)-x)]

cot x = tan[(π/2)-x)] and tax x = cot[(π/2)-x)]

csc x = sec[(π/2)-x)] and sec x = csc[(π/2)-x)]


Composite Argument Properties for Cosine, Sine, and Tangent

cos (A - B) = cos A cos B + sin A sin B

cos (A + B) = cos A cos B - sin A sin B

sin (A - B) = sin A cos B - cos A sin B

sin (A + B) = sin A cos B + cos A sin B

tan (A - B) = (tan A - tan B)/(1 + tan A tan B)

tan (A + B) = (tan A + tan B)/(1 - tan A tan B)

(I know that's a lot to remember, but these are really important!)

Here is Mr. French's device to help you remember them:

Cosine is the "weird" function that is counterintuitive, so think opposite. When you see a "-" on one side, you will need a "+" on the other side (and vice versa). Sine and tangent make sense. A "-" will translate to a "-" on the other side, as will a "+." You just have to remember that in the sine equation, sin comes first, not cos.

Example Problem - Solving an Equation Using the Composite Properties

Links
Drawings, Explanations, and Practice

Personalization :)
Proof:

How do you prove in three steps that a sheet of paper is a lazy dog?
1. A sheet of paper is an ink-lined plane.
2. An inclined plane is a slope up.
3. A slow pup is a lazy dog.

Edward, you're next!