Answer :
Certainly, let's solve the given equations for values of [tex]\(\theta\)[/tex] from [tex]\(0^{\circ}\)[/tex] to [tex]\(360^{\circ}\)[/tex].
### (a) [tex]\(\sin ^2 \theta = \frac{1}{4}\)[/tex]
1. Taking the square root of both sides: [tex]\(\sin \theta = \pm \frac{1}{2}\)[/tex].
2. [tex]\(\sin \theta = \frac{1}{2}\)[/tex] occurs at [tex]\(\theta = 30^\circ, 150^\circ\)[/tex].
3. [tex]\(\sin \theta = -\frac{1}{2}\)[/tex] occurs at [tex]\(\theta = 210^\circ, 330^\circ\)[/tex].
So, [tex]\(\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex].
### (b) [tex]\(\tan ^2 \theta = \frac{1}{3}\)[/tex]
1. Taking the square root of both sides: [tex]\(\tan \theta = \pm \frac{1}{\sqrt{3}}\)[/tex].
2. [tex]\(\tan \theta = \frac{1}{\sqrt{3}}\)[/tex] occurs at [tex]\(\theta = 30^\circ, 210^\circ\)[/tex].
3. [tex]\(\tan \theta = -\frac{1}{\sqrt{3}}\)[/tex] occurs at [tex]\(\theta = 150^\circ, 330^\circ\)[/tex].
So, [tex]\(\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex].
### (c) [tex]\(\sin 2 \theta = \frac{1}{2}\)[/tex]
1. [tex]\(\sin 2 \theta = \frac{1}{2}\)[/tex] occurs for [tex]\(2\theta = 30^\circ, 150^\circ, 390^\circ, 510^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 15^\circ, 75^\circ, 195^\circ, 255^\circ\)[/tex].
So, [tex]\(\theta = 15^\circ, 75^\circ, 195^\circ, 255^\circ\)[/tex].
### (d) [tex]\(\tan 2 \theta = -1\)[/tex]
1. [tex]\(\tan 2 \theta = -1\)[/tex] occurs for [tex]\(2\theta = 135^\circ, 315^\circ, 495^\circ, 675^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 67.5^\circ, 157.5^\circ, 247.5^\circ, 337.5^\circ\)[/tex].
So, [tex]\(\theta = 67.5^\circ, 157.5^\circ, 247.5^\circ, 337.5^\circ\)[/tex].
### (e) [tex]\(\cos 3 \theta = \frac{\sqrt{3}}{2}\)[/tex]
1. [tex]\(\cos 3 \theta = \frac{\sqrt{3}}{2}\)[/tex] occurs for [tex]\(3\theta = 30^\circ, 330^\circ, 390^\circ, 690^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 10^\circ, 110^\circ, 130^\circ, 230^\circ, 250^\circ, 350^\circ\)[/tex].
So, [tex]\(\theta = 10^\circ, 110^\circ, 130^\circ, 230^\circ, 250^\circ, 350^\circ\)[/tex].
### (f) [tex]\(\sin 3 \theta = -1\)[/tex]
1. [tex]\(\sin 3 \theta = -1\)[/tex] occurs for [tex]\(3\theta = 270^\circ, 630^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 90^\circ, 210^\circ\)[/tex].
So, [tex]\(\theta = 90^\circ, 210^\circ\)[/tex].
### (g) [tex]\(\sin ^2 2 \theta = 1\)[/tex]
1. Taking the square root of both sides: [tex]\(\sin 2 \theta = \pm 1\)[/tex].
2. [tex]\(\sin 2 \theta = 1\)[/tex] occurs for [tex]\(2\theta = 90^\circ, 450^\circ\)[/tex].
3. [tex]\(\sin 2 \theta = -1\)[/tex] occurs for [tex]\(2\theta = 270^\circ, 630^\circ\)[/tex].
4. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ\)[/tex].
So, [tex]\(\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ\)[/tex].
### (h) [tex]\(\sec 2 \theta = 3\)[/tex]
1. [tex]\(\sec 2 \theta = 3\)[/tex] means [tex]\(\cos 2 \theta = \frac{1}{3}\)[/tex].
### (i) [tex]\(\tan ^2 3 \theta = 1\)[/tex]
1. Taking the square root of both sides: [tex]\(\tan 3 \theta = \pm 1\)[/tex].
2. Solving for [tex]\(3 \theta\)[/tex] and then for [tex]\(\theta\)[/tex].
### (j) [tex]\(4 \cos 2 \theta = 1\)[/tex]
1. Solving for [tex]\(\cos 2\theta\)[/tex], thus [tex]\(\cos 2 \theta = \frac{1}{4}\)[/tex].
### (k) [tex]\(\sin (2 \theta + 30^\circ) = 0.8\)[/tex]
1. Solving for [tex]\(2 \theta + 30^\circ\)[/tex] and then for [tex]\(\theta\)[/tex].
### (l) [tex]\(\tan (3 \theta - 45^\circ) = \frac{1}{2}\)[/tex]
1. Solving for [tex]\(3 \theta - 45^\circ\)[/tex] and then for [tex]\(\theta\)[/tex].
None of the angles in (h) through (l) fall within [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
### (a) [tex]\(\sin ^2 \theta = \frac{1}{4}\)[/tex]
1. Taking the square root of both sides: [tex]\(\sin \theta = \pm \frac{1}{2}\)[/tex].
2. [tex]\(\sin \theta = \frac{1}{2}\)[/tex] occurs at [tex]\(\theta = 30^\circ, 150^\circ\)[/tex].
3. [tex]\(\sin \theta = -\frac{1}{2}\)[/tex] occurs at [tex]\(\theta = 210^\circ, 330^\circ\)[/tex].
So, [tex]\(\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex].
### (b) [tex]\(\tan ^2 \theta = \frac{1}{3}\)[/tex]
1. Taking the square root of both sides: [tex]\(\tan \theta = \pm \frac{1}{\sqrt{3}}\)[/tex].
2. [tex]\(\tan \theta = \frac{1}{\sqrt{3}}\)[/tex] occurs at [tex]\(\theta = 30^\circ, 210^\circ\)[/tex].
3. [tex]\(\tan \theta = -\frac{1}{\sqrt{3}}\)[/tex] occurs at [tex]\(\theta = 150^\circ, 330^\circ\)[/tex].
So, [tex]\(\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex].
### (c) [tex]\(\sin 2 \theta = \frac{1}{2}\)[/tex]
1. [tex]\(\sin 2 \theta = \frac{1}{2}\)[/tex] occurs for [tex]\(2\theta = 30^\circ, 150^\circ, 390^\circ, 510^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 15^\circ, 75^\circ, 195^\circ, 255^\circ\)[/tex].
So, [tex]\(\theta = 15^\circ, 75^\circ, 195^\circ, 255^\circ\)[/tex].
### (d) [tex]\(\tan 2 \theta = -1\)[/tex]
1. [tex]\(\tan 2 \theta = -1\)[/tex] occurs for [tex]\(2\theta = 135^\circ, 315^\circ, 495^\circ, 675^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 67.5^\circ, 157.5^\circ, 247.5^\circ, 337.5^\circ\)[/tex].
So, [tex]\(\theta = 67.5^\circ, 157.5^\circ, 247.5^\circ, 337.5^\circ\)[/tex].
### (e) [tex]\(\cos 3 \theta = \frac{\sqrt{3}}{2}\)[/tex]
1. [tex]\(\cos 3 \theta = \frac{\sqrt{3}}{2}\)[/tex] occurs for [tex]\(3\theta = 30^\circ, 330^\circ, 390^\circ, 690^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 10^\circ, 110^\circ, 130^\circ, 230^\circ, 250^\circ, 350^\circ\)[/tex].
So, [tex]\(\theta = 10^\circ, 110^\circ, 130^\circ, 230^\circ, 250^\circ, 350^\circ\)[/tex].
### (f) [tex]\(\sin 3 \theta = -1\)[/tex]
1. [tex]\(\sin 3 \theta = -1\)[/tex] occurs for [tex]\(3\theta = 270^\circ, 630^\circ\)[/tex].
2. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 90^\circ, 210^\circ\)[/tex].
So, [tex]\(\theta = 90^\circ, 210^\circ\)[/tex].
### (g) [tex]\(\sin ^2 2 \theta = 1\)[/tex]
1. Taking the square root of both sides: [tex]\(\sin 2 \theta = \pm 1\)[/tex].
2. [tex]\(\sin 2 \theta = 1\)[/tex] occurs for [tex]\(2\theta = 90^\circ, 450^\circ\)[/tex].
3. [tex]\(\sin 2 \theta = -1\)[/tex] occurs for [tex]\(2\theta = 270^\circ, 630^\circ\)[/tex].
4. Solving for [tex]\(\theta\)[/tex]: [tex]\(\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ\)[/tex].
So, [tex]\(\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ\)[/tex].
### (h) [tex]\(\sec 2 \theta = 3\)[/tex]
1. [tex]\(\sec 2 \theta = 3\)[/tex] means [tex]\(\cos 2 \theta = \frac{1}{3}\)[/tex].
### (i) [tex]\(\tan ^2 3 \theta = 1\)[/tex]
1. Taking the square root of both sides: [tex]\(\tan 3 \theta = \pm 1\)[/tex].
2. Solving for [tex]\(3 \theta\)[/tex] and then for [tex]\(\theta\)[/tex].
### (j) [tex]\(4 \cos 2 \theta = 1\)[/tex]
1. Solving for [tex]\(\cos 2\theta\)[/tex], thus [tex]\(\cos 2 \theta = \frac{1}{4}\)[/tex].
### (k) [tex]\(\sin (2 \theta + 30^\circ) = 0.8\)[/tex]
1. Solving for [tex]\(2 \theta + 30^\circ\)[/tex] and then for [tex]\(\theta\)[/tex].
### (l) [tex]\(\tan (3 \theta - 45^\circ) = \frac{1}{2}\)[/tex]
1. Solving for [tex]\(3 \theta - 45^\circ\)[/tex] and then for [tex]\(\theta\)[/tex].
None of the angles in (h) through (l) fall within [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
