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) = cos

^{2}x - sin

^{2}x

Using Pythagorean Identity, this can also be rewritten...

[sin

^{2}x = 1 - cos

^{2}x ] [cos

^{2}x = 1-sin

^{2}x]

cos(2x) = cos

^{2}x - (1-cos

^{2}x)

cos(2x) =2cos

^{2}x - 1

-OR-

cos(2x) = (1-sin

^{2}x) - sin

^{2}x

cos (2x) = 1 - 2sin

^{2}x

Example 1: Rewrite using Double Angle Formula

a.

cos (6x)

=cos [2(3x)]

=cos

^{2}x (3x) - sin

^{2}(3x)

-OR -

=2cos

^{2}x (3x) - 1

-OR-

=1-2sin

^{2}x (3x)

b.

sin (0.5x) [¼ + ¼

= 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 | =
= 1/cosx |

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