Join IDNLearn.com today and start getting the answers you've been searching for. Our Q&A platform offers detailed and trustworthy answers to ensure you have the information you need.
Sagot :
To find which of the given options is correct for [tex]\(\theta\)[/tex], we need to analyze the values of [tex]\(\sin \theta\)[/tex], [tex]\(\cos \theta\)[/tex], [tex]\(\sec \theta\)[/tex], and [tex]\(\tan \theta\)[/tex] when [tex]\(\sin \theta = -\frac{2}{3}\)[/tex].
1. Determine [tex]\(\cos \theta\)[/tex]:
Using the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Given that [tex]\(\sin \theta = -\frac{2}{3}\)[/tex],
[tex]\[ \left( -\frac{2}{3} \right)^2 + \cos^2 \theta = 1 \][/tex]
Simplify the square of [tex]\(\sin \theta\)[/tex]:
[tex]\[ \frac{4}{9} + \cos^2 \theta = 1 \][/tex]
Subtract [tex]\(\frac{4}{9}\)[/tex] from both sides:
[tex]\[ \cos^2 \theta = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9} \][/tex]
Therefore, [tex]\(\cos \theta\)[/tex] can be:
[tex]\[ \cos \theta = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3} \][/tex]
2. Determine [tex]\(\tan \theta\)[/tex]:
Using the definition [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex], we have two possibilities for [tex]\(\tan \theta\)[/tex] based on the values of [tex]\(\cos \theta\)[/tex]:
If [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{2}{3}}{\frac{\sqrt{5}}{3}} = -\frac{2}{\sqrt{5}} \][/tex]
If [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}} = \frac{2}{\sqrt{5}} \][/tex]
Now, we’ll check each option to see which of these conditions match:
- Option A: [tex]\(\sec \theta=\frac{3}{\sqrt{5}}\)[/tex] and [tex]\(\tan \theta=-\frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{5}}{3}} = \frac{3}{\sqrt{5}} \][/tex]
[tex]\( \cos \theta = \frac{\sqrt{5}}{3} \)[/tex] results in:
[tex]\[ \tan \theta = -\frac{2}{\sqrt{5}} \][/tex]
Option A matches both conditions for [tex]\(\theta\)[/tex] when [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex].
- Option B: [tex]\(\sec \theta=-\frac{3}{2}\)[/tex] and [tex]\(\tan \theta=\frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
If [tex]\(\sec \theta = -\frac{3}{2}\)[/tex], then [tex]\(\cos \theta = -\frac{2}{3}\)[/tex]. This does not match our previous calculations where [tex]\(\cos \theta\)[/tex] should be [tex]\(\pm \frac{\sqrt{5}}{3}\)[/tex]. Therefore, Option B is incorrect.
- Option C: [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\tan \theta = \frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \cos \theta = \frac{\sqrt{5}}{3} \][/tex]
When [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex], [tex]\(\tan \theta\)[/tex] should be [tex]\( -\frac{2}{\sqrt{5}} \)[/tex]. Therefore, Option C is incorrect because of the mismatch in the [tex]\(\tan \theta\)[/tex] value.
- Option D: [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\tan \theta = \frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \cos \theta = -\frac{\sqrt{5}}{3} \][/tex]
This implies:
[tex]\[ \tan \theta = \frac{2}{\sqrt{5}} \][/tex]
This condition is satisfied. Therefore, Option D is also correct for this value.
Hence, the possible answers are:
- [tex]\(\boxed{\text{A and D}}\)[/tex]
1. Determine [tex]\(\cos \theta\)[/tex]:
Using the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Given that [tex]\(\sin \theta = -\frac{2}{3}\)[/tex],
[tex]\[ \left( -\frac{2}{3} \right)^2 + \cos^2 \theta = 1 \][/tex]
Simplify the square of [tex]\(\sin \theta\)[/tex]:
[tex]\[ \frac{4}{9} + \cos^2 \theta = 1 \][/tex]
Subtract [tex]\(\frac{4}{9}\)[/tex] from both sides:
[tex]\[ \cos^2 \theta = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9} \][/tex]
Therefore, [tex]\(\cos \theta\)[/tex] can be:
[tex]\[ \cos \theta = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3} \][/tex]
2. Determine [tex]\(\tan \theta\)[/tex]:
Using the definition [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex], we have two possibilities for [tex]\(\tan \theta\)[/tex] based on the values of [tex]\(\cos \theta\)[/tex]:
If [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{2}{3}}{\frac{\sqrt{5}}{3}} = -\frac{2}{\sqrt{5}} \][/tex]
If [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}} = \frac{2}{\sqrt{5}} \][/tex]
Now, we’ll check each option to see which of these conditions match:
- Option A: [tex]\(\sec \theta=\frac{3}{\sqrt{5}}\)[/tex] and [tex]\(\tan \theta=-\frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{5}}{3}} = \frac{3}{\sqrt{5}} \][/tex]
[tex]\( \cos \theta = \frac{\sqrt{5}}{3} \)[/tex] results in:
[tex]\[ \tan \theta = -\frac{2}{\sqrt{5}} \][/tex]
Option A matches both conditions for [tex]\(\theta\)[/tex] when [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex].
- Option B: [tex]\(\sec \theta=-\frac{3}{2}\)[/tex] and [tex]\(\tan \theta=\frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
If [tex]\(\sec \theta = -\frac{3}{2}\)[/tex], then [tex]\(\cos \theta = -\frac{2}{3}\)[/tex]. This does not match our previous calculations where [tex]\(\cos \theta\)[/tex] should be [tex]\(\pm \frac{\sqrt{5}}{3}\)[/tex]. Therefore, Option B is incorrect.
- Option C: [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\tan \theta = \frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \cos \theta = \frac{\sqrt{5}}{3} \][/tex]
When [tex]\(\cos \theta = \frac{\sqrt{5}}{3}\)[/tex], [tex]\(\tan \theta\)[/tex] should be [tex]\( -\frac{2}{\sqrt{5}} \)[/tex]. Therefore, Option C is incorrect because of the mismatch in the [tex]\(\tan \theta\)[/tex] value.
- Option D: [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\tan \theta = \frac{2}{\sqrt{5}}\)[/tex]
[tex]\[ \cos \theta = -\frac{\sqrt{5}}{3} \][/tex]
This implies:
[tex]\[ \tan \theta = \frac{2}{\sqrt{5}} \][/tex]
This condition is satisfied. Therefore, Option D is also correct for this value.
Hence, the possible answers are:
- [tex]\(\boxed{\text{A and D}}\)[/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.
