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 θ cos θ [Rewrite as a multiple of 2]

6 sin θ cos θ = 3 · [2 sinθ cos θ ] > [Sine Double Angle Formula! :o]

6 sin θ cos θ = 3 sin 2 θ


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