2. no calculator. find each angle $\theta$ with the given trigonometric value. use the domain $0^{circ} leq…

Explanation:

Step1: Recall the unit - circle values

We know that on the unit circle, (\cos\theta=x) and (\sin\theta = y). The reference angles and their trigonometric values: (\cos30^{\circ}=\frac{\sqrt{3}}{2}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\sin90^{\circ} = 1)

Step2: Solve for (\theta) in (\cos\theta=\frac{\sqrt{3}}{2})

Since (\cos\theta=\frac{\sqrt{3}}{2}), and (\cos\theta) is positive in the first and fourth quadrants. The reference angle (\theta_{r}=30^{\circ}) For the first - quadrant: (\theta = 30^{\circ}) For the fourth - quadrant: (\theta=360^{\circ}-30^{\circ}=330^{\circ})

Step3: Solve for (\theta) in (\sin\theta=\frac{-\sqrt{2}}{2})

Since (\sin\theta) is negative, it is in the third and fourth quadrants. The reference angle (\theta_{r} = 45^{\circ}) For the third - quadrant: (\theta=180^{\circ}+45^{\circ}=225^{\circ}) For the fourth - quadrant: (\theta=360^{\circ}-45^{\circ}=315^{\circ})

Step4: Solve for (\theta) in (\sin\theta = 1)

We know that (\sin90^{\circ}=1). In the domain (0^{\circ}\leq\theta<360^{\circ}), (\theta = 90^{\circ})

Answer:

a. (\theta = 30^{\circ}) or (\theta=330^{\circ}) b. (\theta = 225^{\circ}) or (\theta=315^{\circ}) c. (\theta = 90^{\circ})