103 Trigonometry Problems

204 103 Trigonometry Problems
Lagrange’s Interpolation Formula
Let x 0 ,x 1 ,...,x n be distinct real numbers, and let y 0 ,y 1 ,...,y n be arbitrary real
numbers. Then there exists a unique polynomial P(x)of degree at most n such that
P(x i ) = y i , i = 0, 1,...,n. This polynomial is given by
P(x) =
n∑
i=0
y i (x − x 0 ) ···(x − x i−1 )(x − x i+1 ) ···(x − x n )
(x i − x 0 ) ···(x i − x i−1 )(x i − x i+1 ) ···(x i − x n ) .
Law of Cosines
In a triangle ABC,
|CA| 2 =|AB| 2 +|BC| 2 − 2|AB|·|BC| cos ̸
ABC,
and analogous equations hold for |AB| 2 and |BC| 2 .
Median formula
This is also called the length of the median formula. Let AM be a median in triangle
ABC. Then
|AM| 2 = 2|AB|2 + 2|AC| 2 −|BC| 2
.
4
Minimal Polynomial
We call a polynomial p(x) with integer coefficients irreducible if p(x) cannot be
written as a product of two polynomials with integer coefficients neither of which
is a constant. Suppose that the number α is a root of a polynomial q(x) with integer
coefficients.Among all polynomials with integer coefficients with leading coefficient
1 (i.e., monic polynomials with integer coefficients) that have α as a root, there is
one of smallest degree. This polynomial is the minimal polynomial of α. Let p(x)
denote this polynomial. Then p(x) is irreducible, and for any other polynomial q(x)
with integer coefficients such that q(α) = 0, the polynomial p(x) divides q(x); that
is, q(x) = p(x)h(x) for some polynomial h(x) with integer coefficients.
Orthocenter of a Triangle
The point of intersection of the altitudes.