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# The Trigonometric Functions ## 1.1 Angles ### Degree Measure The measure of an angle is determined by the amount of rotation about the vertex required to move the initial side to the terminal side. One way to measure angles is in degrees. A complete revolution about the vertex is $360^\circ$. T...

# The Trigonometric Functions ## 1.1 Angles ### Degree Measure The measure of an angle is determined by the amount of rotation about the vertex required to move the initial side to the terminal side. One way to measure angles is in degrees. A complete revolution about the vertex is $360^\circ$. Therefore, one degree is $\frac{1}{360}$ of a complete revolution. **Figure 1** shows several angles. Notice that angles are labeled with Greek letters such as $\alpha$ (alpha), $\beta$ (beta), and $\theta$ (theta). In **Figure 1(c)**, notice that the angle $\alpha$ is negative, because the rotation from the initial side to the terminal side is clockwise. \begin{center} \begin{tabular}{ccc} \includegraphics[max width=0.3\textwidth]{image1.png} & \includegraphics[max width=0.3\textwidth]{image2.png} & \includegraphics[max width=0.3\textwidth]{image3.png} \\ (a) & (b) & (c) \end{tabular} \textbf{Figure 1} \end{center} ### Radian Measure To define trigonometric functions of real numbers, we use radian measure. *Definition of Radian Measure*: One radian is the measure of the central angle $\theta$ that intercepts an arc $s$ equal in length to the radius $r$ of the circle. See **Figure 2(a)**. **Figure 2** \begin{center} \begin{tabular}{cc} \includegraphics[max width=0.4\textwidth]{image4.png} & \includegraphics[max width=0.4\textwidth]{image5.png} \\ (a) & (b) \end{tabular} \end{center} Because the circumference of a circle is $2\pi r$, a complete revolution corresponds to an angle of $2\pi$ radians. A straight angle (half revolution) corresponds to an angle of $\pi$ radians. Because $360^\circ = 2\pi$ radians, we obtain $\qquad 180^\circ = \pi \text{ radians}$ and we have the following conversion rules: 1. To convert degrees to radians, multiply degrees by $\frac{\pi}{180}$. 2. To convert radians to degrees, multiply radians by $\frac{180}{\pi}$. To apply these two conversion rules, use the basic relationship $\pi \text{ radians} = 180^\circ$. See **Figure 2(b)**. ## Example 1 Converting Between Degrees and Radians (a) Convert $60^\circ$ to radians. (b) Convert $\frac{\pi}{4}$ radians to degrees. ### Solution (a) $60^\circ = 60 \text{ degrees } (\frac{\pi \text{ radians }}{180 \text{ degrees}}) = \frac{\pi}{3} \text{ radians}$ (b) $\frac{\pi}{4} \text{ radians } = (\frac{\pi}{4} \text{ radians }) (\frac{180 \text{ degrees }}{\pi \text{ radians}}) = 45^\circ$ ## Angles Two angles that have the same initial and terminal sides are coterminal.