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

Question

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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!
The equivalent angles within $$[0, 2π)$$ are: - (a) $$ \frac{π}{5} $$ - (b) $$ \frac{2π}{3} $$ - (c) $$ \frac{π}{2} $$ - (d) $$ \frac{5π}{3} $$

(a)
(b)
©
(d)

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