Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. ―203° 20'

Verified step by step guidance

1. Convert the given angle from degrees and minutes to decimal degrees. To do this, divide the minutes by 60 and add to the degrees: \(-203^\circ 20' = -203^\circ - \frac{20}{60}^\circ\).
2. Calculate the equivalent positive angle by adding 360° to the converted angle until the result is positive. This is because coterminal angles differ by full rotations of 360°.
3. If the result from the previous step is still negative, add another 360° to ensure the angle is positive.
4. Once you have a positive angle, check if it is less than 360°. If it is greater than 360°, subtract 360° to find the least positive coterminal angle.
5. Express the final angle in degrees and minutes if required, by converting the decimal part back to minutes.

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position, differing by full rotations of 360 degrees (or 2π radians). To find a coterminal angle, you can add or subtract multiples of 360° from the given angle. For example, if you have an angle of 30°, adding 360° gives you another coterminal angle of 390°.

Degrees and Minutes

In trigonometry, angles can be measured in degrees and minutes, where one degree is divided into 60 minutes. This notation is particularly useful for expressing angles with greater precision. For instance, an angle of -203° 20' means -203 degrees and 20 minutes, which can be converted to a decimal format for easier calculations.

Finding the Least Positive Angle

To find the least positive angle coterminal with a given angle, you need to adjust the angle into the range of 0° to 360°. This is done by adding 360° until the angle is positive. For example, if you start with -203° 20', you would add 360° to find the equivalent positive angle, ensuring it is the smallest positive measure that is coterminal.