Answered

Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Check all that apply. If [tex] \csc \theta = \frac{13}{5} [/tex], then:

A. [tex] \sec \theta = \frac{13}{5} [/tex]
B. [tex] \tan \theta = \frac{5}{12} [/tex]
C. [tex] \sin \theta = \frac{5}{13} [/tex]
D. [tex] \cos \theta = \frac{5}{13} [/tex]

Sagot :

GinnyAnswer
Let's analyze the given information and go through the solution step-by-step to determine if each statement is true or false.

Given:
[tex]\[ \csc \theta = \frac{13}{5} \][/tex]

We know that [tex]\(\csc \theta\)[/tex] is the reciprocal of [tex]\(\sin \theta\)[/tex]. Therefore,
[tex]\[ \sin \theta = \frac{1}{\csc \theta} = \frac{5}{13} \][/tex]

Next, we use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]

Substituting [tex]\(\sin \theta = \frac{5}{13}\)[/tex],
[tex]\[ \left( \frac{5}{13} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{25}{169} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{25}{169} \][/tex]
[tex]\[ \cos^2 \theta = \frac{169}{169} - \frac{25}{169} \][/tex]
[tex]\[ \cos^2 \theta = \frac{144}{169} \][/tex]
[tex]\[ \cos \theta = \sqrt{\frac{144}{169}} \][/tex]
[tex]\[ \cos \theta = \frac{12}{13} \][/tex]

Now that we have [tex]\(\cos \theta\)[/tex], we can find [tex]\(\sec \theta\)[/tex], [tex]\(\tan \theta\)[/tex], [tex]\(\sin \theta\)[/tex], and [tex]\(\cos \theta\)[/tex]:

1. [tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex]:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12} \][/tex]

2. [tex]\(\tan \theta\)[/tex] is given by:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12} \][/tex]

3. We already have [tex]\(\sin \theta\)[/tex]:
[tex]\[ \sin \theta = \frac{5}{13} \][/tex]

4. We already have [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \frac{12}{13} \][/tex]

Now let's validate each of the given conditions:

A. [tex]\(\sec \theta = \frac{5}{13}\)[/tex]
- We found [tex]\(\sec \theta = \frac{13}{12}\)[/tex].
- Therefore, this statement is false.

B. [tex]\(\tan \theta = \frac{5}{12}\)[/tex]
- We found [tex]\(\tan \theta = \frac{5}{12}\)[/tex].
- Therefore, this statement is true.

C. [tex]\(\sin \theta = \frac{5}{13}\)[/tex]
- We found [tex]\(\sin \theta = \frac{5}{13}\)[/tex].
- Therefore, this statement is true.

D. [tex]\(\cos \theta = \frac{5}{13}\)[/tex]
- We found [tex]\(\cos \theta = \frac{12}{13}\)[/tex].
- Therefore, this statement is false.

So the truth values are:
- A: False
- B: True
- C: True
- D: False
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.