## Wednesday, November 25, 2009

### Period 1: 4.5 Prove Trigonometric Identities, Part 1

Hello 8)

So... What are trigonometric identities?
Trig. identities are equations that are true for all angles.
A counter example is proving that a trigonometric identity is not true.

Today, we learned about the Double Angle Formula; Used to express double an angle.
So how are these equations formed ?

Recall ;
sin (a+b) = sin a cos b + cos a sin b
|> Set a equal to b. This is similar to saying "double a"
Let x represent our new variables to represent a new equation

sin (x+x) = sin x cos x + cos x sin x

This can be rewritten as:

sin(2x) = 2(sin x cos x)

Recall;
cos (a+b) = cos a cos b + sin a sin b
|> Set a equal to b. This is similar to saying "double a"

cos (x+x) = cos x cos x + sin x sin x

This can rewritten as:

cos(2x) = cos2 x - sin2 x

Using Pythagorean Identity, this can also be rewritten...
[sin2 x = 1 - cos2 x ] [cos2 x = 1-sin2 x]

cos(2x) = cos2 x - (1-cos2 x)
cos(2x) =2cos2 x - 1

-OR-

cos(2x) = (1-sin2 x) - sin2 x
cos (2x) = 1 - 2sin2 x

Example 1: Rewrite using Double Angle Formula
a.
cos (6x)
=cos [2(3x)]
=cos2 x (3x) - sin2 (3x)
-OR -
=2cos2 x (3x) - 1
-OR-
=1-2sin2 x (3x)

b.
sin (0.5x) [¼ + ¼ = ½ ]
= sin [2( ¼ x)]
= 2 sin ( ¼ x) cos ( ¼ x)

Example 2: Express as a single sine/cosine function
6 sin Î¸ cos Î¸ [Rewrite as a multiple of 2]

6 sin Î¸ cos Î¸ = 3 · [2 sinÎ¸ cos Î¸ ] > [Sine Double Angle Formula! :o]

6 sin Î¸ cos Î¸ = 3 sin 2 Î¸

Don't forget how to do a Left Side - Right Side check !

Prove secx = tanx cscx

 LEFT SIDE: secx RIGHT SIDE: tanx cscx =1 / cosx = sinx /cosx x 1/sinx = 1/cosx