Answer :
Sure, let's solve each of the given trigonometric equations for [tex]\(\theta\)[/tex] within the range [tex]\([0^\circ, 360^\circ]\)[/tex].
[tex]\(\text{(a)} \sin^2 \theta = \frac{1}{4}\)[/tex]
For [tex]\(\sin \theta\)[/tex] to satisfy this equation, [tex]\(\sin \theta = \pm \frac{1}{2}\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(30^\circ, 150^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(b)} \tan^2 \theta = \frac{1}{3}\)[/tex]
For [tex]\(\tan \theta\)[/tex] to satisfy this equation, [tex]\(\tan \theta = \pm \frac{1}{\sqrt{3}}\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(30^\circ, 150^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(c)} \sin 2 \theta = \frac{1}{2}\)[/tex]
For [tex]\(\sin 2 \theta\)[/tex] to satisfy this equation, [tex]\(2 \theta = 30^\circ, 150^\circ, 390^\circ,\)[/tex] and [tex]\(510^\circ\)[/tex] (since [tex]\(\sin 2 \theta\)[/tex] is periodic with [tex]\(360^\circ\)[/tex]).
Hence, [tex]\(\theta\)[/tex] can be [tex]\(15^\circ, 75^\circ, 195^\circ,\)[/tex] and [tex]\(255^\circ\)[/tex].
[tex]\(\text{(d)} \tan 2 \theta = -1\)[/tex]
For [tex]\(\tan 2 \theta\)[/tex] to satisfy this equation, [tex]\(2 \theta = 135^\circ, 315^\circ,\)[/tex] [tex]\(495^\circ,\)[/tex] and [tex]\(675^\circ\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(135^\circ\)[/tex] and [tex]\(225^\circ\)[/tex].
[tex]\(\text{(e)} \cos 3 \theta = \frac{\sqrt{3}}{2}\)[/tex]
For [tex]\(\cos 3 \theta\)[/tex] to satisfy this equation, [tex]\(3 \theta = 30^\circ, 330^\circ,\)[/tex] and [tex]\(690^\circ\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(10^\circ, 110^\circ,\)[/tex] and [tex]\(250^\circ\)[/tex].
[tex]\(\text{(f)} \sin 3 \theta = -1\)[/tex]
For [tex]\(\sin 3 \theta\)[/tex] to satisfy this equation, [tex]\(3 \theta = 270^\circ, 630^\circ,\)[/tex] and [tex]\(990^\circ\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(90^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(g)} \sin^2 2 \theta = 1\)[/tex]
For [tex]\(\sin 2 \theta\)[/tex] to satisfy this equation, [tex]\(\sin 2 \theta = \pm 1\)[/tex].
Hence, [tex]\(2 \theta = 90^\circ, 270^\circ, 450^\circ,\)[/tex] and [tex]\(630^\circ\)[/tex], meaning, [tex]\(\theta = 0^\circ, 90^\circ, 180^\circ, 270^\circ,\)[/tex] [tex]\(45^\circ, 135^\circ, 225^\circ,\)[/tex] and [tex]\(315^\circ\)[/tex].
[tex]\(\text{(h)} \sec 2 \theta = 3\)[/tex]
For [tex]\(\sec 2 \theta\)[/tex] to satisfy this equation, [tex]\(\cos 2 \theta = \frac{1}{3}\)[/tex].
Hence, [tex]\(2 \theta = 60^\circ, 300^\circ,\)[/tex] [tex]\(420^\circ,\)[/tex] and [tex]\(660^\circ\)[/tex], meaning, [tex]\(\theta = 20^\circ, 160^\circ, 200^\circ,\)[/tex] and [tex]\(340^\circ\)[/tex].
[tex]\(\text{(i)} \tan^2 3 \theta = 1\)[/tex]
For [tex]\(\tan 3 \theta\)[/tex] to satisfy this equation, [tex]\(\tan 3 \theta = \pm 1\)[/tex].
Hence, [tex]\(3 \theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ, 405^\circ,\)[/tex] and [tex]\(495^\circ\)[/tex], meaning, [tex]\(\theta = 15^\circ, 75^\circ, 135^\circ, 195^\circ, 255^\circ,\)[/tex] and [tex]\(315^\circ\)[/tex].
[tex]\(\text{(j)} 4 \cos 2 \theta = 1\)[/tex]
For [tex]\(\cos 2 \theta\)[/tex] to satisfy this equation, [tex]\(\cos 2 \theta = \frac{1}{4}\)[/tex].
Hence, [tex]\(2 \theta = 60^\circ, 300^\circ,\)[/tex] [tex]\(420^\circ,\)[/tex] meaning, [tex]\(\theta = 30^\circ, 150^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(k)} \sin \left(2 \theta + 30^\circ \right) = 0.8\)[/tex]
For [tex]\(\sin \left(2 \theta + 30^\circ \right)\)[/tex] to satisfy this equation, [tex]\(2 \theta + 30^\circ = 53.13^\circ, 126.87^\circ,\)[/tex] [tex]\(413.13^\circ,\)[/tex] and [tex]\(486.87^\circ\)[/tex], meaning, [tex]\(\theta \approx 20^\circ, 80^\circ, 200^\circ,\)[/tex] and [tex]\(260^\circ\)[/tex].
[tex]\(\text{(l)} \tan \left(3 \theta - 45^\circ \right) = \frac{1}{2}\)[/tex]
For [tex]\(\tan \left(3 \theta - 45^\circ \right)\)[/tex] to satisfy this equation, [tex]\(3 \theta - 45^\circ = 26.57^\circ, 206.57^\circ,\)[/tex] [tex]\(386.57^\circ,\)[/tex] and [tex]\(566.57^\circ\)[/tex], meaning, [tex]\(\theta \approx 21^\circ, 81^\circ, 201^\circ,\)[/tex] and [tex]\(261^\circ\)[/tex].
Therefore, the following angles (in degrees) solve the equations:
1. [tex]\((a) 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex]
2. [tex]\((b) 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex]
3. [tex]\((c) 15^\circ, 75^\circ, 195^\circ, 255^\circ\)[/tex]
4. [tex]\((d) 135^\circ, 225^\circ\)[/tex]
5. [tex]\((e) 10^\circ, 110^\circ, 250^\circ\)[/tex]
6. [tex]\((f) 90^\circ, 210^\circ, 330^\circ\)[/tex]
7. [tex]\((g) 0^\circ, 90^\circ, 180^\circ, 270^\circ, 45^\circ, 135^\circ, 225^\circ, 315^\circ\)[/tex]
8. [tex]\((h) 20^\circ, 160^\circ, 200^\circ, 340^\circ\)[/tex]
9. [tex]\((i) 15^\circ, 75^\circ, 135^\circ, 195^\circ, 255^\circ, 315^\circ\)[/tex]
10. [tex]\((j) 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex]
11. [tex]\((k) 20^\circ, 80^\circ, 200^\circ, 260^\circ\)[/tex]
12. [tex]\((l) 21^\circ, 81^\circ, 201^\circ, 261^\circ\)[/tex]
[tex]\(\text{(a)} \sin^2 \theta = \frac{1}{4}\)[/tex]
For [tex]\(\sin \theta\)[/tex] to satisfy this equation, [tex]\(\sin \theta = \pm \frac{1}{2}\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(30^\circ, 150^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(b)} \tan^2 \theta = \frac{1}{3}\)[/tex]
For [tex]\(\tan \theta\)[/tex] to satisfy this equation, [tex]\(\tan \theta = \pm \frac{1}{\sqrt{3}}\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(30^\circ, 150^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(c)} \sin 2 \theta = \frac{1}{2}\)[/tex]
For [tex]\(\sin 2 \theta\)[/tex] to satisfy this equation, [tex]\(2 \theta = 30^\circ, 150^\circ, 390^\circ,\)[/tex] and [tex]\(510^\circ\)[/tex] (since [tex]\(\sin 2 \theta\)[/tex] is periodic with [tex]\(360^\circ\)[/tex]).
Hence, [tex]\(\theta\)[/tex] can be [tex]\(15^\circ, 75^\circ, 195^\circ,\)[/tex] and [tex]\(255^\circ\)[/tex].
[tex]\(\text{(d)} \tan 2 \theta = -1\)[/tex]
For [tex]\(\tan 2 \theta\)[/tex] to satisfy this equation, [tex]\(2 \theta = 135^\circ, 315^\circ,\)[/tex] [tex]\(495^\circ,\)[/tex] and [tex]\(675^\circ\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(135^\circ\)[/tex] and [tex]\(225^\circ\)[/tex].
[tex]\(\text{(e)} \cos 3 \theta = \frac{\sqrt{3}}{2}\)[/tex]
For [tex]\(\cos 3 \theta\)[/tex] to satisfy this equation, [tex]\(3 \theta = 30^\circ, 330^\circ,\)[/tex] and [tex]\(690^\circ\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(10^\circ, 110^\circ,\)[/tex] and [tex]\(250^\circ\)[/tex].
[tex]\(\text{(f)} \sin 3 \theta = -1\)[/tex]
For [tex]\(\sin 3 \theta\)[/tex] to satisfy this equation, [tex]\(3 \theta = 270^\circ, 630^\circ,\)[/tex] and [tex]\(990^\circ\)[/tex].
Hence, [tex]\(\theta\)[/tex] can be [tex]\(90^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(g)} \sin^2 2 \theta = 1\)[/tex]
For [tex]\(\sin 2 \theta\)[/tex] to satisfy this equation, [tex]\(\sin 2 \theta = \pm 1\)[/tex].
Hence, [tex]\(2 \theta = 90^\circ, 270^\circ, 450^\circ,\)[/tex] and [tex]\(630^\circ\)[/tex], meaning, [tex]\(\theta = 0^\circ, 90^\circ, 180^\circ, 270^\circ,\)[/tex] [tex]\(45^\circ, 135^\circ, 225^\circ,\)[/tex] and [tex]\(315^\circ\)[/tex].
[tex]\(\text{(h)} \sec 2 \theta = 3\)[/tex]
For [tex]\(\sec 2 \theta\)[/tex] to satisfy this equation, [tex]\(\cos 2 \theta = \frac{1}{3}\)[/tex].
Hence, [tex]\(2 \theta = 60^\circ, 300^\circ,\)[/tex] [tex]\(420^\circ,\)[/tex] and [tex]\(660^\circ\)[/tex], meaning, [tex]\(\theta = 20^\circ, 160^\circ, 200^\circ,\)[/tex] and [tex]\(340^\circ\)[/tex].
[tex]\(\text{(i)} \tan^2 3 \theta = 1\)[/tex]
For [tex]\(\tan 3 \theta\)[/tex] to satisfy this equation, [tex]\(\tan 3 \theta = \pm 1\)[/tex].
Hence, [tex]\(3 \theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ, 405^\circ,\)[/tex] and [tex]\(495^\circ\)[/tex], meaning, [tex]\(\theta = 15^\circ, 75^\circ, 135^\circ, 195^\circ, 255^\circ,\)[/tex] and [tex]\(315^\circ\)[/tex].
[tex]\(\text{(j)} 4 \cos 2 \theta = 1\)[/tex]
For [tex]\(\cos 2 \theta\)[/tex] to satisfy this equation, [tex]\(\cos 2 \theta = \frac{1}{4}\)[/tex].
Hence, [tex]\(2 \theta = 60^\circ, 300^\circ,\)[/tex] [tex]\(420^\circ,\)[/tex] meaning, [tex]\(\theta = 30^\circ, 150^\circ, 210^\circ,\)[/tex] and [tex]\(330^\circ\)[/tex].
[tex]\(\text{(k)} \sin \left(2 \theta + 30^\circ \right) = 0.8\)[/tex]
For [tex]\(\sin \left(2 \theta + 30^\circ \right)\)[/tex] to satisfy this equation, [tex]\(2 \theta + 30^\circ = 53.13^\circ, 126.87^\circ,\)[/tex] [tex]\(413.13^\circ,\)[/tex] and [tex]\(486.87^\circ\)[/tex], meaning, [tex]\(\theta \approx 20^\circ, 80^\circ, 200^\circ,\)[/tex] and [tex]\(260^\circ\)[/tex].
[tex]\(\text{(l)} \tan \left(3 \theta - 45^\circ \right) = \frac{1}{2}\)[/tex]
For [tex]\(\tan \left(3 \theta - 45^\circ \right)\)[/tex] to satisfy this equation, [tex]\(3 \theta - 45^\circ = 26.57^\circ, 206.57^\circ,\)[/tex] [tex]\(386.57^\circ,\)[/tex] and [tex]\(566.57^\circ\)[/tex], meaning, [tex]\(\theta \approx 21^\circ, 81^\circ, 201^\circ,\)[/tex] and [tex]\(261^\circ\)[/tex].
Therefore, the following angles (in degrees) solve the equations:
1. [tex]\((a) 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex]
2. [tex]\((b) 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex]
3. [tex]\((c) 15^\circ, 75^\circ, 195^\circ, 255^\circ\)[/tex]
4. [tex]\((d) 135^\circ, 225^\circ\)[/tex]
5. [tex]\((e) 10^\circ, 110^\circ, 250^\circ\)[/tex]
6. [tex]\((f) 90^\circ, 210^\circ, 330^\circ\)[/tex]
7. [tex]\((g) 0^\circ, 90^\circ, 180^\circ, 270^\circ, 45^\circ, 135^\circ, 225^\circ, 315^\circ\)[/tex]
8. [tex]\((h) 20^\circ, 160^\circ, 200^\circ, 340^\circ\)[/tex]
9. [tex]\((i) 15^\circ, 75^\circ, 135^\circ, 195^\circ, 255^\circ, 315^\circ\)[/tex]
10. [tex]\((j) 30^\circ, 150^\circ, 210^\circ, 330^\circ\)[/tex]
11. [tex]\((k) 20^\circ, 80^\circ, 200^\circ, 260^\circ\)[/tex]
12. [tex]\((l) 21^\circ, 81^\circ, 201^\circ, 261^\circ\)[/tex]
