Sure! Let's solve the problem step by step.
Given:
[tex]\[ \sin \theta = \frac{1}{10} \][/tex]
We know that in quadrant II:
1. [tex]\(\sin \theta\)[/tex] is positive.
2. [tex]\(\cos \theta\)[/tex] is negative.
3. [tex]\(\tan \theta\)[/tex] is negative.
### a) Calculating [tex]\(\cos \theta\)[/tex]
To find [tex]\(\cos \theta\)[/tex], we can use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute the given value for [tex]\(\sin \theta\)[/tex]:
[tex]\[ \left(\frac{1}{10}\right)^2 + \cos^2 \theta = 1 \][/tex]
Simplify:
[tex]\[ \frac{1}{100} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{1}{100} \][/tex]
[tex]\[ \cos^2 \theta = \frac{100}{100} - \frac{1}{100} \][/tex]
[tex]\[ \cos^2 \theta = \frac{99}{100} \][/tex]
So, [tex]\(\cos \theta\)[/tex] is:
[tex]\[ \cos \theta = \pm \sqrt{\frac{99}{100}} \][/tex]
[tex]\[ \cos \theta = \pm \frac{\sqrt{99}}{10} \][/tex]
Since we are in quadrant II where [tex]\(\cos \theta\)[/tex] is negative:
[tex]\[ \cos \theta = -\frac{\sqrt{99}}{10} \][/tex]
We can simplify [tex]\(\sqrt{99}\)[/tex] further as [tex]\(\sqrt{99} = \sqrt{9 \times 11} = 3\sqrt{11}\)[/tex], but for simplicity, let's stick with:
[tex]\[ \cos \theta \approx -0.99498743710662 \][/tex]
### b) Calculating [tex]\(\tan \theta\)[/tex]
Next, we calculate [tex]\(\tan \theta\)[/tex] using:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
[tex]\[ \tan \theta = \frac{\frac{1}{10}}{-\frac{\sqrt{99}}{10}} \][/tex]
[tex]\[ \tan \theta = \frac{1}{-\sqrt{99}} \][/tex]
Rationalize the denominator:
[tex]\[ \tan \theta = -\frac{1}{\sqrt{99}} \times \frac{\sqrt{99}}{\sqrt{99}} \][/tex]
[tex]\[ \tan \theta = -\frac{\sqrt{99}}{99} \][/tex]
So,
[tex]\[ \tan \theta \approx -0.10050378152592121 \][/tex]
### c) Calculating [tex]\(\sec \theta\)[/tex], [tex]\(\csc \theta\)[/tex], and [tex]\(\cot \theta\)[/tex]
The secant function is the reciprocal of the cosine function:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
[tex]\[ \sec \theta = \frac{1}{-\frac{\sqrt{99}}{10}} \][/tex]
[tex]\[ \sec \theta = -\frac{10}{\sqrt{99}} \][/tex]
Rationalize the denominator:
[tex]\[ \sec \theta = -\frac{10 \cdot \sqrt{99}}{99} \][/tex]
[tex]\[ \sec \theta \approx -1.005037815259212 \][/tex]
The cosecant function is the reciprocal of the sine function:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
[tex]\[ \csc \theta = \frac{1}{\frac{1}{10}} = 10 \][/tex]
The cotangent function is the reciprocal of the tangent function:
[tex]\[ \cot \theta = \frac{1}{\tan \theta} \][/tex]
[tex]\[ \cot \theta = \frac{1}{-\frac{\sqrt{99}}{99}} \][/tex]
[tex]\[ \cot \theta = -\frac{99}{\sqrt{99}} = -\sqrt{99} \approx -9.9498743710662 \][/tex]
### Summary
- [tex]\(\cos \theta \approx -0.99498743710662\)[/tex]
- [tex]\(\tan \theta \approx -0.10050378152592121\)[/tex]
- [tex]\(\sec \theta \approx -1.005037815259212\)[/tex]
- [tex]\(\csc \theta = 10\)[/tex]
- [tex]\(\cot \theta \approx -9.9498743710662\)[/tex]
These are the exact and computed values for the trigonometric functions based on the given [tex]\(\sin \theta\)[/tex] and the properties of the quadrant.
