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Section A.14 Highschool Material You Should be Able to Derive

  • Graphs of \(\csc\theta, \sec \theta\) and \(\cot \theta\text{:}\)
\begin{equation*} \csc \theta \end{equation*}
\begin{equation*} \sec \theta \end{equation*}
\begin{equation*} \cot \theta \end{equation*}
  • More Pythagoras
    \begin{align*} \sin^2\theta + \cos^2 \theta &=1 & \xmapsto{\text{divide by $\cos^2\theta$}}&& \tan^2\theta + 1 &= \sec^2\theta\\ \sin^2\theta + \cos^2 \theta &=1 & \xmapsto{\text{divide by $\sin^2\theta$}}&& 1 + \cot^2 \theta &=\csc^2\theta \end{align*}
  • Sine — double angle (set \(\beta =\alpha\) in sine angle addition formula)
    \begin{align*} \sin(2\alpha) &= 2\sin(\alpha)\cos(\alpha) \end{align*}
  • Cosine — double angle (set \(\beta =\alpha\) in cosine angle addition formula)
    \begin{align*} \cos(2\alpha) &= \cos^2(\alpha) - \sin^2(\alpha)\\ &= 2\cos^2(\alpha) - 1 & \text{(use $\sin^2(\alpha)= 1-\cos^2(\alpha)$)}\\ &= 1 - 2\sin^2(\alpha) & \text{(use $\cos^2(\alpha)= 1-\sin^2(\alpha)$)} \end{align*}
  • Composition of trigonometric and inverse trigonometric functions:
    \begin{align*} \cos( \arcsin x) &= \sqrt{1-x^2} & \sec( \arctan x) &= \sqrt{1+x^2} \end{align*}
    and similar expressions.