Question

1. determine the measure of an angle coterminal with each angle in the domain $0^{circ} leq \theta<360^{circ}$ or $0 leq \theta<2 pi$. draw a diagram showing the quadrant in which each angle terminates.

a) 6.75
b) $400^{circ}$
c) $-3$
d) $-105^{circ}$

Explanation:

Step1: Recall the formula for coterminal angles

For angles in radians, if \(\theta\) is an angle, then \(\theta + 2k\pi\) (\(k\in\mathbb{Z}\)) are coterminal angles. For angles in degrees, \(\theta + 360^{\circ}k\) (\(k\in\mathbb{Z}\)) are coterminal angles.

Step2: Solve for part (a)

Given \(\theta = 6.75\) radians.
Since \(2\pi\approx6.28\), \(6.75-2\pi=6.75 - 6.28=0.47\) radians.

Step3: Solve for part (b)

Given \(\theta = 400^{\circ}\).
\(400^{\circ}-360^{\circ}=40^{\circ}\)

Step4: Solve for part (c)

Given \(\theta=- 3\) radians.
\(-3 + 2\pi=-3+6.28 = 3.28\) radians

Step5: Solve for part (d)

Given \(\theta=-105^{\circ}\).
\(-105^{\circ}+360^{\circ}=255^{\circ}\)

Answer:

a) \(0.47\) radians (in the first quadrant)
b) \(40^{\circ}\) (in the first quadrant)
c) \(3.28\) radians (in the second quadrant)
d) \(255^{\circ}\) (in the third quadrant)