Further Trig Identities and Approximations

Compound Angle Identities

Sin(A+B) = SinA CosB + CosA SinB. This is the identity for the sine of two angles added together, A and B.
Cos(A+B) = CosA CosB – SinA SinB. This identity is used for the cosine of two angles added together. Notice that instead of a plus sign in the middle like in the sine compound angle identity, there is a minus sign.
Tan(A+B) = (TanA + TanB) / (1 - TanA TanB). The tangent of two angles added together uses this identity, and is derived using the sine and cosine compound angle identities.

Double Angle Identities

Sin(2A) = 2SinA CosA. This is known as the double angle identity for sine. This is derived from the sine compound angle identity by setting B = A.
Cos(2A) = Cos²A - Sin²A = 2Cos²A - 1 = 1 - 2Sin²A. The double angle identity for cosine has three forms. The other two forms are derived from the primary form using the Pythagorean identity Sin²A + Cos²A = 1.
Tan(2A) = 2TanA / (1 - Tan²A). The double angle identity for tangent is derived using the sine and cosine double angle identities.

Trig Approximations

Sin(x) approximation: For small x, sin(x) ~ x.
Cos(x) approximation: For small x, cos(x) ~ 1 - x²/2.
Tan(x) approximation: For small x, tan(x) ~ x.

Using these identities and approximations will allow for quicker and more efficient solving of various trigonometric problems and equations. Note that these approximations are only accurate for small values of x, and lose accuracy as x increases.