To solve the problem, we need to determine the quadrant in which the terminal side of each angle lies. Here is how to approach each case:

Background

• Quadrants:

  1. Quadrant I: $0 < \theta < \frac{\pi}{2}$
  2. Quadrant II: $\frac{\pi}{2} < \theta < \pi$
  3. Quadrant III: $\pi < \theta < \frac{3\pi}{2}$
  4. Quadrant IV: $\frac{3\pi}{2} < \theta < 2\pi$

• For angles greater than $2\pi$ or negative angles, reduce the angle modulo $2\pi$ (find its coterminal angle within the range $0 \leq \theta < 2\pi$).

Problem Breakdown

(a) $\theta = \frac{8\pi}{3}$

• Reduce modulo $2\pi$:
  $\frac{8\pi}{3} - 2\pi = \frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3}.$ $\frac{2\pi}{3}$ lies in Quadrant II because $\frac{\pi}{2} < \frac{2\pi}{3} < \pi$.

(b) $\theta = \frac{3\pi}{4}$

• $\frac{3\pi}{4}$ is already in the standard range $0 \leq \theta < 2\pi$.
  It lies in Quadrant II because $\frac{\pi}{2} < \frac{3\pi}{4} < \pi$.

(c) $\theta = \frac{7\pi}{6}$

• $\frac{7\pi}{6}$ is in the standard range $0 \leq \theta < 2\pi$.
  It lies in Quadrant III because $\pi < \frac{7\pi}{6} < \frac{3\pi}{2}$.

(d) $\theta = 3$

• Approximate $3$ in radians relative to $\pi$:
  $3 < \pi$, and since $\pi \approx 3.14$, $3$ lies in Quadrant II.

Final Answers

• (a) Quadrant II
• (b) Quadrant II
• (c) Quadrant III
• (d) Quadrant II

Let me know if you would like detailed steps or more explanations!

Follow-up Questions

1. What are coterminal angles, and how are they calculated?
2. How does the quadrant of an angle affect trigonometric functions like sine and cosine?
3. How can you determine the quadrant of an angle from its sine or cosine values?
4. What is the relationship between degrees and radians, and how do you convert between them?
5. What happens when angles are negative? How do they correspond to quadrants?

Tip

To quickly find the quadrant of an angle in radians, reduce it modulo $2\pi$ and compare it to key angles like $\frac{\pi}{2}, \pi, \frac{3\pi}{2},$ and $2\pi$.