Get expert insights and reliable answers to your questions on IDNLearn.com. Find the information you need quickly and easily with our comprehensive and accurate Q&A 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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.
