Problems

In this problem set, we will use angle brackets to denote points, e.g. \(\pt10,\) and square brackets to denote vectors, e.g. \(\vec32.\) This is not standard notation. Typically both are denoted by square brackets (or parentheses), and distinguished by context (if at all).

  1. We reduced the problem of computing \(\RotP\theta AB\) to the problem of computing \(\RotVparen\theta {B-A},\) and found a formula for that, but we never explicitly wrote down the resulting formula for \(\RotP\theta AB.\) So suppose that \(A = \pt {a_1}{a_2} \) and \(B = \pt {b_1}{b_2}, \) and write down an explicit formula for \(\RotP\theta AB. \)
  2. In the video, we established the formula \[ \begin{equation}\label{rot} \RotV\theta{\vec xy} = x\vec{\cos\theta}{\sin\theta} + y\vec{-\sin\theta}{\phantom{\mathop-}\cos\theta} = \vec{x\cos\theta - y\sin\theta}{x\sin\theta + y\cos\theta} \end{equation} \] for rotating a vector \(\vec xy\) by an angle \(\theta\) about the origin. Use this to derive the "angle-sum identities" \[ \begin{equation}\label{angle-sum} \vec {\cos(\alpha+\beta)} {\sin(\alpha+\beta)} = \vec{\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)} {\cos(\alpha)\sin(\beta) + \sin(\alpha)\cos(\beta)}. \end{equation} \]

  3. Establish the "double-angle" identities \[ \begin{align*} \cos(2\theta) &= \cos^2(\theta) - \sin^2(\theta)\\ &= 2\cos^2(\theta) - 1\\ & = 1 - 2\sin^2(\theta)\\ \sin(2\theta) &= 2\cos(\theta)\sin(\theta) \end{align*} \] and the "half-angle" identities \[ \begin{align} \cos(\theta/2) &= \pm\sqrt{\frac{1+\cos(\theta)}2} \label{half-cos}\\ \sin(\theta/2) &= \pm\sqrt{\frac{1-\cos(\theta)}2}. \label{half-sin} \end{align} \]

  4. Find \(\cos15^\circ\), \(\sin15^\circ\), \(\cos75^\circ\), \(\sin75^\circ.\)