Math 12x : Precalculus

Cos Double Angle Formula: cos(2θ) = cos²θ-sin²θ
Sin Double Angle Formula: sin(2θ) = 2sinθcosθ

[\begin{matrix} cos2θ & -sin2θ \\ sin2θ & cos2θ \end{matrix}]

[\begin{matrix} cosθ & -sinθ \\ sinθ & cosθ \end{matrix}]

[\begin{matrix} cosθ & -sinθ \\ sinθ & cosθ \end{matrix}]

[\begin{matrix} cos²θ-sin²θ & -2sinθcosθ \\ 2sinθcosθ & cos²θ-sin²θ \end{matrix}]

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

[\begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}

|\begin{matrix} 5 \\ 12 \end{matrix}]

→

Row Echelon

[\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}

EA=U

|\begin{matrix} 5 \\ 2 \end{matrix}]

→ y = 2 → x = -1

Gauss-Jordan Elimination

Original

[\begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}

|\begin{matrix} 5 \\ 12 \end{matrix}]

→

Row Echelon

[\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}

EA=U

|\begin{matrix} 5 \\ 2 \end{matrix}]

→

Reduced Echelon

[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}

A⁻¹A=I

|\begin{matrix} -1 \\ 2 \end{matrix}]

A⁻¹b=x

→ x = -1, y = 2

Original

[\begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}

|\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

→

Row Echelon

[\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}

EA=U

|\begin{matrix} 1 & 0 \\ -2 & 1 \end{matrix}]

EI=E

→

Reduced Echelon

[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}

A⁻¹A=I

|\begin{matrix} 7 & -3 \\ -2 & 1 \end{matrix}]

A⁻¹I=A⁻¹

→

Application

[\begin{matrix} 7 & -3 \\ -2 & 1 \end{matrix}]

A^-1

[\begin{matrix} 5 \\ 12 \end{matrix}]

[\begin{matrix} -1 \\ 2 \end{matrix}]