MATH 144: Calculus for the Physical Sciences I

Subsection 1.4.1 Things to know

• To convert angles between degrees and radians: $180^o = \pi \text{ radians}\text{.}$
• A point $(x,y)$ on the unit circle at angle $\theta$ with respect to the positive side of the $x$-axis (positive angle meaning counterclockwise rotation) has coordinates $(x,y) = (\cos \theta, \sin \theta)\text{.}$
• Given a right triangle,
  $$\begin{equation*} \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}. \end{equation*}$$
• The four other trig functions can be obtained from $\sin \theta$ and $\cos \theta$ as:
  $$\begin{equation*} \tan \theta = \frac{\sin \theta}{\cos \theta}, \qquad \cot \theta = \frac{\cos \theta}{\sin \theta}, \qquad \sec \theta = \frac{1}{\cos \theta}, \qquad \csc \theta = \frac{1}{\sin \theta}. \end{equation*}$$

• Some useful trigonometric identities:

  • $$\begin{gather*} \sin^2 A + \cos^2 A = 1, \qquad \tan^2 A + 1 = \sec^2 A,\\ 1 + \cot^2 A = \csc^2 A. \end{gather*}$$
    Note that you only need to remember the first one, you can derive the other two from it.
  • $$\begin{gather*} \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B,\\ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B. \end{gather*}$$
  • $$\begin{gather*} \sin 2 A = 2 \sin A \cos A,\\ \cos 2 A = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A. \end{gather*}$$
  • $$\begin{equation*} \sin^2 A = \frac{1}{2} (1 - \cos 2 A), \qquad \cos^2 A = \frac{1}{2} (1 + \cos 2 A ). \end{equation*}$$

Subsection 1.4.2 Review videos