Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve this problem, we need to determine the values of [tex]\(\sec \theta\)[/tex] and [tex]\(\cot \theta\)[/tex] given that [tex]\(\csc \theta = \frac{3}{2}\)[/tex] and [tex]\(\sec \theta < 0\)[/tex].
### Step-by-Step Solution:
1. Find [tex]\(\sin \theta\)[/tex]:
The cosecant function [tex]\(\csc \theta\)[/tex] is defined as the reciprocal of the sine function:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
Given [tex]\(\csc \theta = \frac{3}{2}\)[/tex], we can find [tex]\(\sin \theta\)[/tex]:
[tex]\[ \sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \][/tex]
2. Find [tex]\(\cos \theta\)[/tex]:
Using the Pythagorean identity, which states:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute [tex]\(\sin \theta = \frac{2}{3}\)[/tex]:
[tex]\[ \left(\frac{2}{3}\right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{4}{9} + \cos^2 \theta = 1 \][/tex]
Solving for [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9} \][/tex]
[tex]\[ \cos \theta = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3} \][/tex]
Given that [tex]\(\sec \theta < 0\)[/tex], [tex]\(\cos \theta\)[/tex] must be negative:
[tex]\[ \cos \theta = -\frac{\sqrt{5}}{3} \][/tex]
3. Find [tex]\(\sec \theta\)[/tex]:
The secant function [tex]\(\sec \theta\)[/tex] is defined as the reciprocal of the cosine function:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
Substitute [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[ \sec \theta = \frac{1}{-\frac{\sqrt{5}}{3}} = -\frac{3}{\sqrt{5}} = -\frac{3\sqrt{5}}{5} \][/tex]
Simplifying, we find:
[tex]\[ \sec \theta \approx -1.3416 \][/tex]
4. Find [tex]\(\cot \theta\)[/tex]:
The cotangent function [tex]\(\cot \theta\)[/tex] is defined as the ratio of the cosine function to the sine function:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]
Substitute [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\sin \theta = \frac{2}{3}\)[/tex]:
[tex]\[ \cot \theta = \frac{-\frac{\sqrt{5}}{3}}{\frac{2}{3}} = -\frac{\sqrt{5}}{2} \][/tex]
Simplifying, we find:
[tex]\[ \cot \theta \approx -1.1180 \][/tex]
### Final Answer
Given that [tex]\(\csc \theta = \frac{3}{2}\)[/tex] and [tex]\(\sec \theta < 0\)[/tex], we have found:
[tex]\[ \sec \theta \approx -1.3416 \quad \text{and} \quad \cot \theta \approx -1.1180 \][/tex]
### Step-by-Step Solution:
1. Find [tex]\(\sin \theta\)[/tex]:
The cosecant function [tex]\(\csc \theta\)[/tex] is defined as the reciprocal of the sine function:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
Given [tex]\(\csc \theta = \frac{3}{2}\)[/tex], we can find [tex]\(\sin \theta\)[/tex]:
[tex]\[ \sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \][/tex]
2. Find [tex]\(\cos \theta\)[/tex]:
Using the Pythagorean identity, which states:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute [tex]\(\sin \theta = \frac{2}{3}\)[/tex]:
[tex]\[ \left(\frac{2}{3}\right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{4}{9} + \cos^2 \theta = 1 \][/tex]
Solving for [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9} \][/tex]
[tex]\[ \cos \theta = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3} \][/tex]
Given that [tex]\(\sec \theta < 0\)[/tex], [tex]\(\cos \theta\)[/tex] must be negative:
[tex]\[ \cos \theta = -\frac{\sqrt{5}}{3} \][/tex]
3. Find [tex]\(\sec \theta\)[/tex]:
The secant function [tex]\(\sec \theta\)[/tex] is defined as the reciprocal of the cosine function:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
Substitute [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[ \sec \theta = \frac{1}{-\frac{\sqrt{5}}{3}} = -\frac{3}{\sqrt{5}} = -\frac{3\sqrt{5}}{5} \][/tex]
Simplifying, we find:
[tex]\[ \sec \theta \approx -1.3416 \][/tex]
4. Find [tex]\(\cot \theta\)[/tex]:
The cotangent function [tex]\(\cot \theta\)[/tex] is defined as the ratio of the cosine function to the sine function:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]
Substitute [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\sin \theta = \frac{2}{3}\)[/tex]:
[tex]\[ \cot \theta = \frac{-\frac{\sqrt{5}}{3}}{\frac{2}{3}} = -\frac{\sqrt{5}}{2} \][/tex]
Simplifying, we find:
[tex]\[ \cot \theta \approx -1.1180 \][/tex]
### Final Answer
Given that [tex]\(\csc \theta = \frac{3}{2}\)[/tex] and [tex]\(\sec \theta < 0\)[/tex], we have found:
[tex]\[ \sec \theta \approx -1.3416 \quad \text{and} \quad \cot \theta \approx -1.1180 \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
