Question

for each angle below, determine the quadrant in which the terminal side of the angle is found and find the corresponding reference angle \\(\hat{\theta}\\).

a. \\(\theta = \dfrac{10\pi}{3}\\) is found in quadrant select an answer \\(\vee\\) and \\(\hat{\theta} = \square\\)
b. \\(\theta = -\dfrac{5\pi}{4}\\) is found in quadrant select an answer \\(\vee\\) and \\(\hat{\theta} = \square\\)
c. \\(\theta = \dfrac{19\pi}{6}\\) is found in quadrant select an answer \\(\vee\\) and \\(\hat{\theta} = \square\\)

Explanation:

Step1: Simplify θ=10π/3

Subtract 2π (full rotation): 10π/3 - 2π = 10π/3 - 6π/3 = 4π/3.

Step2: Find quadrant for 4π/3

4π/3 is between π (3π/3) and 3π/2 (9π/6), so Quadrant III.

Step3: Calculate reference angle for 4π/3

Ref angle = θ - π = 4π/3 - π = π/3.

Step4: Simplify θ=-5π/4

Add 2π: -5π/4 + 8π/4 = 3π/4.

Step5: Find quadrant for 3π/4

3π/4 is between π/2 (2π/4) and π (4π/4), so Quadrant II.

Step6: Calculate reference angle for 3π/4

Ref angle = π - θ = π - 3π/4 = π/4.

Step7: Simplify θ=19π/6

Subtract 2π (12π/6): 19π/6 - 12π/6 = 7π/6.

Step8: Find quadrant for 7π/6

7π/6 is between π (6π/6) and 3π/2 (9π/6), so Quadrant III.

Step9: Calculate reference angle for7π/6

Ref angle = θ - π =7π/6 - π= π/6.

Answer:

a. Quadrant III, $\frac{\pi}{3}$
b. Quadrant II, $\frac{\pi}{4}$
c. Quadrant III, $\frac{\pi}{6}$