In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle.17𝜋5

Verified step by step guidance

1. 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 (multiples of \$2\pi$ radians) from the given angle.
2. Given the angle \(\frac{17\pi}{5}\), 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{17\pi}{5}\( until the result is less than \)2\pi$. This involves finding a common denominator and performing the subtraction.
4. Express \$2\pi\( with a denominator of 5: \)2\pi = \frac{10\pi}{5}\(. Subtract \)\frac{10\pi}{5}\( from \)\frac{17\pi}{5}$ to find the coterminal angle.
5. Continue subtracting \(\frac{10\pi}{5}\) if necessary until the angle is less than \$2\pi\(. The resulting angle will be the positive coterminal angle less than \)2\pi$.

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 450 degrees, subtracting 360 degrees gives you a coterminal angle of 90 degrees.

Radians and Degrees

Radians and degrees are two units for measuring angles. One full rotation (360 degrees) is equivalent to 2π radians. To convert between these units, you can use the relationships: degrees = radians × (180/π) and radians = degrees × (π/180). Understanding this conversion is essential for working with angles in trigonometry.

Finding Positive Angles

When tasked with finding a positive angle less than a specified value, you typically need to ensure the angle is within the range of 0 to 360 degrees (or 0 to 2π radians). If the angle exceeds this range, you can subtract 360 degrees (or 2π radians) until the angle falls within the desired interval. This process is crucial for determining the correct coterminal angle.