In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle.25𝜋6

Verified step by step guidance

1. First, understand that coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle, you can add or subtract full rotations (in radians, this is 2\pi) from the given angle.
2. Given the angle \( \frac{25\pi}{6} \), we need to find a positive angle less than \( 2\pi \) that is coterminal with it.
3. Calculate the equivalent angle by subtracting \( 2\pi \) from \( \frac{25\pi}{6} \) until the result is less than \( 2\pi \).
4. Convert \( 2\pi \) to a fraction with a denominator of 6 to make subtraction easier: \( 2\pi = \frac{12\pi}{6} \).
5. Subtract \( \frac{12\pi}{6} \) from \( \frac{25\pi}{6} \) to find the coterminal angle: \( \frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6} \).

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle, you can add or subtract multiples of 360 degrees (or 2π radians) from the given angle. For example, if you have an angle of 25π/6, you can subtract 2π to find a positive angle that is coterminal.

Radians and Degrees

Angles can be measured in radians or degrees, with 360 degrees equivalent to 2π radians. Understanding the conversion between these two units is essential for solving problems in trigonometry. In this case, since the angle is given in radians, it is important to work within that system to find coterminal angles.

Finding Positive Angles

When tasked with finding a positive angle less than a specified value, such as 360 degrees or 2π radians, it is crucial to ensure the resulting angle is within the desired range. This often involves subtracting full rotations (360 degrees or 2π radians) from the original angle until the result falls within the specified limits.