Prep Precalculus D 2006-07

Chapter 5 Section 6: Double and Half Argument Properties

So it's almost Christmas and what's a better gift than the Double and Half Argument Properties? I can't think of one. Enjoy! In this section we take a step beyond the composite argument property. When you write cos2x as cos(x + x) the composite argument property is used. A double argument property can result by expressing cos2x in terms of sines and cosines of x. Half argument properties can also be derived from the double argument properties.

PRODUCTS and SQUARES of COSINE and SINE
sin x cos x = 1/2 sin2x - Product of sine and cosine property
cos²x = 1/2 cos2x - Square of cosine property
sin²x=1/2-1/2 cos 2x - Square of sine property.

Example 1:
sin30 cos30 = 1/2 sin60 => 1/2 (root(3)/2) => root(3)/4
sin30 cos30 => 1/2 (root(3)/2) => root(3)/4

DOUBLE ARGUMENT PROPERTIES
This is how the double argument property for cosine is derived.
cos2x = cos(x + x)
= cos x cos x - sin x sin x
= cos²x - sin²x
- Double Argument for Sine: MEMORIZE
sin2A = 2sinA cosA
- Double Argument for Cosine: MEMORIZE
cos2A = cos²A - sin²A
cos2A = 2 cos²A -1
cos2A = 1 - 2 sin²A
- Double Argument Property for Tangent
tan2A = (2 tanA) ÷ (1 - tan²A)

HALF ARGUMENT PROPERTIES
- Half Argument Property for Sine
sin(1/2)A = +/- root(1/2(1-cosA))
-Half Argument Property for Cosine
cos(1/2)A = +/- root(1/2(1+cosA))
-Half Argument Property for Tangent
tan(1/2)A = +/- root((1-cosA)÷(1+cosA)) = (1 - cosA)÷(sinA) = (sinA) ÷ (1 + cos A)

Example 2:
Express the sin100 in terms of cos200.
First look for a connection between the two numbers.
100 is half of 200, so the Half Properties will be used.
Determine what quadrant the first term (sin100) falls into. It falls into quadrant II in this case, and the +/- before the square root becomes positive, since sin is positive in quadrant II. Finally plug the numbers into the equation.
sin100 = + root((1/2)(1-cos200))

Here are some additional links:
http://www.andrews.edu/~calkins/math/webtexts/numb18.htm

http://www.keymath.com/x7111.xml

So I could'nt get the picture I wanted into the blog, so here's a good quote. Two men were walking down the streets of New York. One was a busy businessman. The other man said to the businessman, "Wow, the crickets sound beautiful today." The businessman replied, "How can you possibly hear the crickets with all the noise around? All the taxis and people block out anything else." The first man responded, "You hear what you listen for."