Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To solve the problem given that [tex]\(\cot{\theta} = \frac{3}{4}\)[/tex] and the angle [tex]\(\theta\)[/tex] is in the third quadrant, let’s proceed step-by-step:
1. Identify relevant trigonometric relationships:
[tex]\(\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \frac{3}{4}\)[/tex]
2. Analyze the quadrant information:
Since [tex]\(\theta\)[/tex] is in the third quadrant, both [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex] are negative.
3. Determine [tex]\(\tan{\theta}\)[/tex]:
[tex]\[ \tan{\theta} = \frac{1}{\cot{\theta}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \][/tex]
In the third quadrant, [tex]\(\tan{\theta}\)[/tex] is positive since both the sine and cosine are negative, and their quotient gives a positive value. Therefore:
[tex]\[ \tan{\theta} = \frac{4}{3} \][/tex]
4. Express [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex] in terms of a common variable:
[tex]\[ \cot{\theta} = \frac{3}{4} = \frac{\cos{\theta}}{\sin{\theta}} \implies \cos{\theta} = 3k \text{ and } \sin{\theta} = 4k \][/tex]
5. Use the Pythagorean identity:
The Pythagorean identity states:
[tex]\[ \sin^2{\theta} + \cos^2{\theta} = 1 \][/tex]
Substituting [tex]\(\cos{\theta} = 3k\)[/tex] and [tex]\(\sin{\theta} = 4k\)[/tex]:
[tex]\[ (3k)^2 + (4k)^2 = 1 \][/tex]
[tex]\[ 9k^2 + 16k^2 = 1 \][/tex]
[tex]\[ 25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5} \][/tex]
6. Find [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex]:
[tex]\[ \sin{\theta} = 4k = 4 \times \frac{1}{5} = \frac{4}{5} \][/tex]
[tex]\[ \cos{\theta} = 3k = 3 \times \frac{1}{5} = \frac{3}{5} \][/tex]
Since both sine and cosine are negative in the third quadrant:
[tex]\[ \sin{\theta} = -\frac{4}{5} \quad \text{and} \quad \cos{\theta} = -\frac{3}{5} \][/tex]
7. Determine [tex]\(\csc{\theta}\)[/tex]:
[tex]\[ \csc{\theta} = \frac{1}{\sin{\theta}} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \][/tex]
Now compare these values to the options given:
A. [tex]\(\sin{\theta} = \frac{3}{5}\)[/tex] [tex]\(\rightarrow\)[/tex] Incorrect, it should be [tex]\(-\frac{4}{5}\)[/tex].
B. [tex]\(\csc{\theta} = -\frac{5}{3}\)[/tex] [tex]\(\rightarrow\)[/tex] Incorrect, it should be [tex]\(-\frac{5}{4}\)[/tex].
C. [tex]\(\cos{\theta} = -\frac{3}{5}\)[/tex] [tex]\(\rightarrow\)[/tex] Correct.
D. [tex]\(\tan{\theta} = \frac{4}{3}\)[/tex] [tex]\(\rightarrow\)[/tex] Correct.
Therefore, the correct answers are C and D.
1. Identify relevant trigonometric relationships:
[tex]\(\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \frac{3}{4}\)[/tex]
2. Analyze the quadrant information:
Since [tex]\(\theta\)[/tex] is in the third quadrant, both [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex] are negative.
3. Determine [tex]\(\tan{\theta}\)[/tex]:
[tex]\[ \tan{\theta} = \frac{1}{\cot{\theta}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \][/tex]
In the third quadrant, [tex]\(\tan{\theta}\)[/tex] is positive since both the sine and cosine are negative, and their quotient gives a positive value. Therefore:
[tex]\[ \tan{\theta} = \frac{4}{3} \][/tex]
4. Express [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex] in terms of a common variable:
[tex]\[ \cot{\theta} = \frac{3}{4} = \frac{\cos{\theta}}{\sin{\theta}} \implies \cos{\theta} = 3k \text{ and } \sin{\theta} = 4k \][/tex]
5. Use the Pythagorean identity:
The Pythagorean identity states:
[tex]\[ \sin^2{\theta} + \cos^2{\theta} = 1 \][/tex]
Substituting [tex]\(\cos{\theta} = 3k\)[/tex] and [tex]\(\sin{\theta} = 4k\)[/tex]:
[tex]\[ (3k)^2 + (4k)^2 = 1 \][/tex]
[tex]\[ 9k^2 + 16k^2 = 1 \][/tex]
[tex]\[ 25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5} \][/tex]
6. Find [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex]:
[tex]\[ \sin{\theta} = 4k = 4 \times \frac{1}{5} = \frac{4}{5} \][/tex]
[tex]\[ \cos{\theta} = 3k = 3 \times \frac{1}{5} = \frac{3}{5} \][/tex]
Since both sine and cosine are negative in the third quadrant:
[tex]\[ \sin{\theta} = -\frac{4}{5} \quad \text{and} \quad \cos{\theta} = -\frac{3}{5} \][/tex]
7. Determine [tex]\(\csc{\theta}\)[/tex]:
[tex]\[ \csc{\theta} = \frac{1}{\sin{\theta}} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \][/tex]
Now compare these values to the options given:
A. [tex]\(\sin{\theta} = \frac{3}{5}\)[/tex] [tex]\(\rightarrow\)[/tex] Incorrect, it should be [tex]\(-\frac{4}{5}\)[/tex].
B. [tex]\(\csc{\theta} = -\frac{5}{3}\)[/tex] [tex]\(\rightarrow\)[/tex] Incorrect, it should be [tex]\(-\frac{5}{4}\)[/tex].
C. [tex]\(\cos{\theta} = -\frac{3}{5}\)[/tex] [tex]\(\rightarrow\)[/tex] Correct.
D. [tex]\(\tan{\theta} = \frac{4}{3}\)[/tex] [tex]\(\rightarrow\)[/tex] Correct.
Therefore, the correct answers are C and D.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
