a.) What quadrant would $$ \frac{3 \pi}{4} $$ end up in? a. I b. II c. III d. IV b.) What would its reference angle be?

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To determine the quadrant and reference angle for $$ \frac{3 \pi}{4} $$, we can follow these steps:

### Part a: Determine the Quadrant

1. **Understanding the angle**: The angle $$ \frac{3 \pi}{4} $$ is in radians. To understand its position, we can convert it to degrees:
   $$
   \text{Degrees} = \frac{3 \pi}{4} \times \frac{180}{\pi} = 135^\circ
   $$

2. **Identifying the quadrant**: 
   - Angles from $$ 0^\circ $$ to $$ 90^\circ $$ are in Quadrant I.
   - Angles from $$ 90^\circ $$ to $$ 180^\circ $$ are in Quadrant II.
   - Angles from $$ 180^\circ $$ to $$ 270^\circ $$ are in Quadrant III.
   - Angles from $$ 270^\circ $$ to $$ 360^\circ $$ are in Quadrant IV.

Since $$ 135^\circ $$ is between $$ 90^\circ $$ and $$ 180^\circ $$, it is in **Quadrant II**.

### Part b: Determine the Reference Angle

1. **Calculating the reference angle**: The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. For angles in Quadrant II, the reference angle $$ \theta' $$ can be calculated as:
   $$
   \theta' = 180^\circ - \theta
   $$
   where $$ \theta $$ is the angle in degrees.

Substituting $$ \theta = 135^\circ $$:
   $$
   \theta' = 180^\circ - 135^\circ = 45^\circ
   $$

2. **Converting back to radians** (if needed):
   $$
   \theta' = 45^\circ \times \frac{\pi}{180} = \frac{\pi}{4}
   $$

### Summary of Answers
- a.) The angle $$ \frac{3 \pi}{4} $$ is in **Quadrant II** (b).
- b.) The reference angle is $$ \frac{\pi}{4} $$.
a.) $$ \frac{3 \pi}{4} $$ is in **Quadrant II**. b.) The reference angle is $$ \frac{\pi}{4} $$.

a.) What quadrant would \( \frac{3 \pi}{4} \) end up in? a. I b. II c. III d. IV b.) What would its reference angle be?

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