Definitions: s i n θ = y r c o s θ = x r t a n θ = y x s i n ( θ + α ) = s i n θ ⋅ c o s α + c o s θ ⋅ s i n α c o s ( θ + α ) = c o s θ ⋅ c o s α − s i n θ ⋅ s i n α Proof: r = x 2 + y 2 θ = t a n − 1 y x x ′ = r ⋅ c o s ( θ + α ) = r ( c o s θ ⋅ c o s α − s i n θ ⋅ s i n α ) = r ( x r c o s α − y r s i n α ) = x ⋅ c o s α − y ⋅ s i n α y ′ = r ⋅ s i n ( θ + α ) = r ( s i n θ ⋅ c o s α + c o s θ ⋅ s i n α ) = r ( y r c o s α + x r s i n α ) = y ⋅ c o s α + x ⋅ s i n α {\displaystyle {\begin{aligned}{\text{Definitions:}}\\sin\theta &={\frac {y}{r}}\\cos\theta &={\frac {x}{r}}\\tan\theta &={\frac {y}{x}}\\sin(\theta +\alpha )&=sin\theta \cdot cos\alpha +cos\theta \cdot sin\alpha \\cos(\theta +\alpha )&=cos\theta \cdot cos\alpha -sin\theta \cdot sin\alpha \\\\{\text{Proof:}}\\r&={\sqrt {x^{2}+y^{2}}}\\\theta &=tan^{-1}{\frac {y}{x}}\\x'&=r\cdot cos(\theta +\alpha )\\&=r(cos\theta \cdot cos\alpha -sin\theta \cdot sin\alpha )\\&=r({\frac {x}{r}}cos\alpha -{\frac {y}{r}}sin\alpha )\\&=x\cdot cos\alpha -y\cdot sin\alpha \\y'&=r\cdot sin(\theta +\alpha )\\&=r(sin\theta \cdot cos\alpha +cos\theta \cdot sin\alpha )\\&=r({\frac {y}{r}}cos\alpha +{\frac {x}{r}}sin\alpha )\\&=y\cdot cos\alpha +x\cdot sin\alpha \\\end{aligned}}}
