# Math 12x : Precalculus

 Cos Double Angle Formula: cos(2θ) = cos²θ-sin²θ Sin Double Angle Formula: sin(2θ) = 2sinθcosθ $\left[\begin{array}{cc}cos2\theta & -sin2\theta \\ sin2\theta & cos2\theta \end{array}\right]$ = $\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]$ $\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]$ = $\left[\begin{array}{cc}cos²\theta -sin²\theta & -2sin\theta cos\theta \\ 2sin\theta cos\theta & cos²\theta -sin²\theta \end{array}\right]$
 Cos Double Angle Formula: cos(2θ) = 2cos²θ-1 Sin Double Angle Formula: sin(2θ) = 2sinθcosθ Pythagorean Iden Formula: cos²θ + sin²θ = 1 2cos²θ - 1 = 2(10.1/2)² - 1 = 102.01/2 - 1 = 100.01/2 = cos2θ 2cosθsinθ = 2(10.1/2)(i0.i/2) = 10.1*i0.i/2 = i00.0i/2 = sin2θ cos²θ+sin²θ = (10.1/2)²+(i0.i/2)² = 102.01/4 + 102.01/4 = 4/4 = 1
Gaussian Elimination
Original
 $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right$ A
 $\left|\begin{array}{c}5\\ 12\end{array}\right]$ b
 →
Row Echelon
 $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right$ EA=U
 $\left|\begin{array}{c}5\\ 2\end{array}\right]$ Eb
 → y = 2 → x = -1
Gauss-Jordan Elimination
Original
 $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right$ A
 $\left|\begin{array}{c}5\\ 12\end{array}\right]$ b
 →
Row Echelon
 $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right$ EA=U
 $\left|\begin{array}{c}5\\ 2\end{array}\right]$ Eb
 →
Reduced Echelon
 $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right$ A⁻¹A=I
 $\left|\begin{array}{c}-1\\ 2\end{array}\right]$ A⁻¹b=x
 → x = -1, y = 2
Original
 $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right$ A
 $\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ I
 →
Row Echelon
 $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right$ EA=U
 $\left|\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]$ EI=E
 →
Reduced Echelon
 $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right$ A⁻¹A=I
 $\left|\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$ A⁻¹I=A⁻¹
 →
Application
 $\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$ A-1
 $\left[\begin{array}{c}5\\ 12\end{array}\right]$ b
 =
 $\left[\begin{array}{c}-1\\ 2\end{array}\right]$ x