Answer :
Given the equation [tex]\(\tan(\theta) = -\sqrt{19}\)[/tex] and that [tex]\(\theta\)[/tex] is in the second quadrant, we aim to find the value of [tex]\(\cos(\theta)\)[/tex]. Here is a step-by-step solution to determine this:
1. Identify the properties of the tangent and cosine functions in quadrants:
- In the second quadrant, [tex]\(\tan(\theta)\)[/tex] is negative, [tex]\(\sin(\theta)\)[/tex] is positive, and [tex]\(\cos(\theta)\)[/tex] is negative.
2. Use the trigonometric identities:
The Pythagorean identity states:
[tex]\[ 1 + \tan^2(\theta) = \sec^2(\theta) \][/tex]
Given [tex]\(\tan(\theta) = -\sqrt{19}\)[/tex]:
[tex]\[ \tan^2(\theta) = (-\sqrt{19})^2 = 19 \][/tex]
[tex]\[ 1 + 19 = \sec^2(\theta) \][/tex]
[tex]\[ \sec^2(\theta) = 20 \][/tex]
3. Relate the secant and cosine functions:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} \][/tex]
Therefore, [tex]\(\sec^2(\theta) = \frac{1}{\cos^2(\theta)}\)[/tex]:
[tex]\[ \frac{1}{\cos^2(\theta)} = 20 \][/tex]
[tex]\[ \cos^2(\theta) = \frac{1}{20} \][/tex]
4. Determine [tex]\(\cos(\theta)\)[/tex]:
Since we are in the second quadrant and [tex]\(\cos(\theta)\)[/tex] is negative:
[tex]\[ \cos(\theta) = -\sqrt{\frac{1}{20}} = -\frac{1}{\sqrt{20}} \][/tex]
Simplifying further:
[tex]\[ \cos(\theta) = -\frac{1}{\sqrt{20}} = -\frac{1}{2\sqrt{5}} = -\frac{\sqrt{5}}{10} \][/tex]
Given:
[tex]\[ \frac{\sqrt{5}\cdot\sqrt{2}}{10\cdot\sqrt{2}} = -\frac{\sqrt{10}}{10\cdot2} = -\frac{\sqrt{10}}{8} = \cos(\theta) \][/tex]
Upon comparing this result with the given options:
A. [tex]\(-\frac{\sqrt{17}}{6}\)[/tex]
B. [tex]\(\frac{\sqrt{17}}{6}\)[/tex]
C. [tex]\(-\frac{\sqrt{12}}{6}\)[/tex]
D. [tex]\(\frac{\sqrt{10}}{8}\)[/tex]
We find that none exactly matches our reduced form [tex]\(\cos(\theta) = -\frac{\sqrt{10}}{8}\)[/tex]. Thus:
None of the calculated options are correct selections as per exact reduction we performed.
* Correction, since this value is consistent with the given cosine evaluation of one:
```
We return [tex]\(\boxed{-\frac{\sqrt{10}}{8}}\)[/tex]
```
1. Identify the properties of the tangent and cosine functions in quadrants:
- In the second quadrant, [tex]\(\tan(\theta)\)[/tex] is negative, [tex]\(\sin(\theta)\)[/tex] is positive, and [tex]\(\cos(\theta)\)[/tex] is negative.
2. Use the trigonometric identities:
The Pythagorean identity states:
[tex]\[ 1 + \tan^2(\theta) = \sec^2(\theta) \][/tex]
Given [tex]\(\tan(\theta) = -\sqrt{19}\)[/tex]:
[tex]\[ \tan^2(\theta) = (-\sqrt{19})^2 = 19 \][/tex]
[tex]\[ 1 + 19 = \sec^2(\theta) \][/tex]
[tex]\[ \sec^2(\theta) = 20 \][/tex]
3. Relate the secant and cosine functions:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} \][/tex]
Therefore, [tex]\(\sec^2(\theta) = \frac{1}{\cos^2(\theta)}\)[/tex]:
[tex]\[ \frac{1}{\cos^2(\theta)} = 20 \][/tex]
[tex]\[ \cos^2(\theta) = \frac{1}{20} \][/tex]
4. Determine [tex]\(\cos(\theta)\)[/tex]:
Since we are in the second quadrant and [tex]\(\cos(\theta)\)[/tex] is negative:
[tex]\[ \cos(\theta) = -\sqrt{\frac{1}{20}} = -\frac{1}{\sqrt{20}} \][/tex]
Simplifying further:
[tex]\[ \cos(\theta) = -\frac{1}{\sqrt{20}} = -\frac{1}{2\sqrt{5}} = -\frac{\sqrt{5}}{10} \][/tex]
Given:
[tex]\[ \frac{\sqrt{5}\cdot\sqrt{2}}{10\cdot\sqrt{2}} = -\frac{\sqrt{10}}{10\cdot2} = -\frac{\sqrt{10}}{8} = \cos(\theta) \][/tex]
Upon comparing this result with the given options:
A. [tex]\(-\frac{\sqrt{17}}{6}\)[/tex]
B. [tex]\(\frac{\sqrt{17}}{6}\)[/tex]
C. [tex]\(-\frac{\sqrt{12}}{6}\)[/tex]
D. [tex]\(\frac{\sqrt{10}}{8}\)[/tex]
We find that none exactly matches our reduced form [tex]\(\cos(\theta) = -\frac{\sqrt{10}}{8}\)[/tex]. Thus:
None of the calculated options are correct selections as per exact reduction we performed.
* Correction, since this value is consistent with the given cosine evaluation of one:
```
We return [tex]\(\boxed{-\frac{\sqrt{10}}{8}}\)[/tex]
```
