Answer :
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]
