# Trigonometric Functions

- Author:
- Michael Horvath

- Topic:
- Trigonometry

*(Move the point labeled "Move" to adjust the angle between the ray and the x-axis.)*

**If Hypoteneuse = r = 1 (a unit circle), then:**Opposite = sin(θ) = y Adjacent = cos(θ) = x

**SOHCAHTOA:**SOH: Opposite / Hypoteneuse = sin(θ) = y / r CAH: Adjacent / Hypoteneuse = cos(θ) = x / r TOA: Opposite / Adjacent = tan(θ) = y / x

**Other Functions:**Hypoteneuse / Opposite = csc(θ) = r / y = 1 / sin(θ) Hypoteneuse / Adjacent = sec(θ) = r / x = 1 / cos(θ) Adjacent / Opposite = cot(θ) = x / y = 1 / tan(θ) 1 = csc(θ)^2 - cot(θ)^2 1 = sec(θ)^2 - tan(θ)^2 1 = sin(θ)^2 + cos(θ)^2 1 = sec(θ) - exsec(θ) 1 = csc(θ) - coexsec(θ) 1 = vers(θ) + cos(θ) 1 = sin(θ) + covers(θ) hav(θ) = vers(θ) / 2

**Pythagorean Theorem:**c^2 = a^2 + b^2 where Adjacent = a, Opposite = b, and Hypoteneuse = c.

**Law of Sines:**2 * r = a / sin(A) = b / sin(B) = c / sin(C) where "r" is the radius of the circumcircle, and a = Adjacent, b = Opposite, and c = Hypoteneuse, and A = the angle opposite a, B = the angle opposite b, and C = the angle opposite c.

**Law of Cosines:**cos(A) = (c^2 + b^2 - a^2) / (2 * b * c) where a = Adjacent, b = Opposite, and c = Hypoteneuse, and A = the angle opposite a.

**Law of Tangents:**(a + b) / (a - b) = tan((A + B) / 2) / tan((A - B) / 2) where a = Adjacent, b = Opposite, and c = Hypoteneuse, and A = the angle opposite a, and B = the angle opposite b.

**Dot Product:**A · B = cos(θ) = x

_{A}* x

_{B}+ y

_{A}* y

_{B }where A and B are vectors with lengths equal to 1.