(a) \( \frac{11 \pi}{5} \) (b) \( \frac{-16 \pi}{3} \) (c) \( \frac{57 \pi}{2} \) (d) \( \frac{15 \pi}{9} \)

It looks like you've provided four angles expressed in terms of π. A common task with such angles is to find their **equivalent angles within the standard interval \([0, 2π)\)**. Let's go through each one step-by-step. ### (a) \( \frac{11π}{5} \) **Standard Interval:** \([0, 2π)\) 1. **Determine how many full rotations (2π) are in the angle:** \[ \frac{11π}{5} = 2π \times \frac{11}{10} = 2π + \frac{π}{5} \] 2. **Subtract the full rotation:** \[ \frac{11π}{5} - 2π = \frac{11π}{5} - \frac{10π}{5} = \frac{π}{5} \] **Equivalent Angle:** \( \frac{π}{5} \) --- ### (b) \( \frac{-16π}{3} \) **Standard Interval:** \([0, 2π)\) 1. **Convert negative angle to positive by adding multiples of 2π:** \[ \frac{-16π}{3} + 6π = \frac{-16π}{3} + \frac{18π}{3} = \frac{2π}{3} \] **Equivalent Angle:** \( \frac{2π}{3} \) --- ### (c) \( \frac{57π}{2} \) **Standard Interval:** \([0, 2π)\) 1. **Determine the number of full rotations:** \[ 2π = \frac{4π}{2} \quad \text{so,} \quad \frac{57π}{2} \div \frac{4π}{2} = 14.25 \text{ rotations} \] 2. **Subtract the full rotations (14 × 2π):** \[ \frac{57π}{2} - 14 \times 2π = \frac{57π}{2} - \frac{56π}{2} = \frac{π}{2} \] **Equivalent Angle:** \( \frac{π}{2} \) --- ### (d) \( \frac{15π}{9} \) **Simplify the Fraction:** \[ \frac{15π}{9} = \frac{5π}{3} \] **Check if it's within \([0, 2π)\):** \[ \frac{5π}{3} \approx 300° \quad \text{which is within} \quad [0, 2π) \] **Equivalent Angle:** \( \frac{5π}{3} \) --- ### Summary of Equivalent Angles within \([0, 2π)\): - **(a) \( \frac{π}{5} \)** - **(b) \( \frac{2π}{3} \)** - **(c) \( \frac{π}{2} \)** - **(d) \( \frac{5π}{3} \)** If you had a different task in mind for these angles, please let me know!