5-3 Composite Argument Properties

This section is all about lots of equations!!

The Odd-Even Properties
Even
cos (x) = cos (-x)
sec (x) = sec (-x)
Odd
sin(x) = -sin (-x)
csc(x) = -csc (-x)
tan (x) = -tan (-x)
cot (x) = -cot (-x)

The Cofunction Properties:
Degrees
cos theta = sin (90 degrees - theta)
sin theta = cos (90 degrees - theta)
cot theta = tan (90 degrees - theta)
tan theta = cot (90 degrees - theta)
csc theta = sec (90 degrees - theta)
sec theta = csc (90 degrees - theta)
Radians
cos x = sin (pi/2 - x)
sin x = cos (pi/2 - x)
cot x = tan (pi/2 - x)
tan x = cot (pi/2 - x)
csc x = sec (pi/2 - x)
sec x = csc (pi/2 - x)

The Composite Argument Properties for Cosine, Sine, and Tangent
Cosine
cos (A-B) = cosAcosB + sinAsinB
cos (A+B) = cosAcosB - sinAsinB
Sine
sin(A-B) = sinAcosB - cosAsinB
sin(A+B) = sinAcosB + cosAsinB
Tangent
tan (A-B) = (tanA - tanB) / (1 + tanAtanB)
tan (A+B) = (tanA + tanB) / (1 - tanAtanB)

You can use these properties to solve trigonometric equations algebraically.

Example Problem:

Find the general solution for x algebraically

cos xcos80 - sinxsin80 = 5

Since the composite argument property for cos(A+B) = cosAcosB - sinAsinB,

cos(x+80) = 0.5
x+80 = arccos 0.5
x = -80 arccos 0.5
x = -80 plus/minus inverse cos (0.5) + 2 pi n
x= 2 pi n - 80 plus/minus 1.047198...


For more help go to http://www.andrews.edu/~calkins/math/webtexts/numb18.htm#COFU
Natalie you're next!

Here's a puzzle that is usually attributed to Einstein, who may or may not have written it.

There are 5 houses in 5 different colours. In each house lives a person of a different nationality. The 5 owners each drink a certain type of beverage, smoke a certain brand of cigar, and keep a certain pet. Using the clues below can you determine who owns the fish?

The Brit lives in a red house.
The Swede keeps dogs as pets.
The Dane drinks tea.
The green house is on the immediate left of the white house.
The green house owner drinks coffee.
The person who smokes Pall Mall rears birds.
The owner of the yellow house smokes Dunhill.
The man living in the house right in the middle drinks milk.
The Norwegian lives in the first house.
The man who smokes Blend lives next door to the one who keeps cats.
The man who keeps horses lives next door to the man who smokes Dunhill.
The owner who smokes Blue Master drinks beer.
The German smokes Prince.
The Norwegian lives next to the blue house.
The man who smokes Blend has a neighbour who drinks water.





...supposedly according to Einstein 98% will not solve this.






The German owns the fish.