Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
