91Ó°ÊÓ

A coterminal angle is an angle that shares the same terminal side with another angle, but may have a different rotation. To find coterminal angles, you can add or subtract 360° or, in radians, add or subtract \(2\pi\). This is because 360° or \(2\pi\) represents a full circle, and rotating a full circle from any angle will end at the same position. For example, let's consider the angle \(\frac{8\pi}{3}\). To find its coterminal angle within the range of 0° to 360°, we subtract 360° (or \(2\pi\)) as many times as needed until the angle lies between 0 and 360 degrees. From the exercise, \( 480° - 360° = 120° \). Thus, \(\frac{8\pi}{3}\) is coterminal with 120°, which is equivalent to \(\frac{2\pi}{3}\). This approach helps in simplifying complex angles and better understanding their positions on the unit circle.

Converting degrees to radians involves a simple formula: multiply the degree measurement by \(\frac{\button{\bm{\bm{\bm{π}}}}}{180°}\). This is because 180 degrees is equivalent to \(\button{\bm{π}}\) radians. For instance, to convert 480° to radians, use the conversion formula: \(480° \times \frac{\button{\bm{\bm{\bm{π}}}}}{180°} = \frac{8\bm{π}}{3}\). Conversely, to convert from radians to degrees, multiply the radian measurement by \(\button{\bm{\bm{\bm{180°}}}}} {\bm{\bm{π}}}\). This conversion process is essential for understanding angles given in different units and for performing calculations that require a specific unit.

Second quadrant angles are angles that lie between 90° and 180°. These angles have distinct characteristics in trigonometry and geometry. For example, their sine values are positive, while cosine values are negative. When finding the reference angle for a second quadrant angle, subtract the angle from 180°. This will give you the smallest angle the terminal side makes with the x-axis. In the exercise, 120° is in the second quadrant. To find its reference angle, subtract it from 180°: \(180° - 120° = 60°\) or in radians, \( \pi - \frac {2\pi} {3} = \frac \pi {3}\). Understanding the quadrant helps in determining the correct reference angle and assists with solving trigonometric problems.