Trig Angle Identities

Fundamental Identity

$\sin^2a + \cos^2a = 1$

Therefore:

$\cos^2a = 1 - \sin^2a$

$\sin^2a = 1 - \cos^2a$

Angle Sum Identity definitions

The Sine of two angles added together identity is as follows:

$\sin(a + b) = \sin(a)\cos(b) + \sin(b)\cos(a)$

And subtraction:

$\sin(a - b) = \sin(a)\cos(b) - \sin(b)\cos(a)$

The Cosine of two angles:

$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$

And subtraction:

$\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$

Cosine Properties

Cosine is an Even Function. Meaning, any negative inputs are equal to the equivalent positive input.

$\cos(-c) = \cos(c)$

We can use the angle sum definition of $a + a$ to defined $\cos(2a)$:

$\cos(2a) = \cos(a + a) = \cos(a)\cos(a) - \sin(a)\sin(a) = \cos^2(a) - \sin^2(a)$

Or simply:

$\cos(2a) = \cos^2(a) - \sin^2(a)$

We can express entirely in terms of $\cos$ Using the Fundemantla Identity:

Since we know

$\sin^2(a) = 1 - \cos^2(a)$

Then

$= \cos^2(a) - (1 - \cos^2(a))$ $= \cos^2(a) - 1 + cos^2(a))$ $= 2\cos^2(a) - 1$ $= \cos(2a)$ (need to understand this part)

Cosine Reduction Identity

$2\cos^2a = \cos 2a + 1$

$\cos^2a = \frac{1}{2} (1 + \cos 2a)$

Sine Properties

Sine is an Odd Function. Meaning, any negative inputs are equal to the negative value of the function.

$\sin(-c) = -\sin(c)$

Sine Reduction Identity

$2\sin^2a + \cos 2a = 1$ $2\sin^2a = 1 - \cos 2a$ $\sin^2a = \frac{1}{2} (1 - \cos 2a)$