Answered

Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

6. If [tex]\cot \theta=\frac{17}{4}[/tex], then find [tex]\frac{(2+2 \sin \theta)(1-\sin \theta)}{(1+\cos \theta)(2-2 \cos \theta)}[/tex].

Sagot :

GinnyAnswer
Certainly! Let's break down the problem step by step to solve the expression given [tex]\(\cot \theta = \frac{17}{4}\)[/tex].

1. Given Information:
[tex]\[ \cot \theta = \frac{17}{4} \][/tex]
Recall that:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]

2. Determine [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:

First, we need to work out [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex].

We know that:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{17}{4} \][/tex]
Therefore,
[tex]\[ \cos \theta = 17k \][/tex]
[tex]\[ \sin \theta = 4k \][/tex]

To find [tex]\(k\)[/tex], we use the Pythagorean identity:
[tex]\[ \cos^2 \theta + \sin^2 \theta = 1 \][/tex]

Substitute [tex]\(\cos \theta = 17k\)[/tex] and [tex]\(\sin \theta = 4k\)[/tex]:
[tex]\[ (17k)^2 + (4k)^2 = 1 \][/tex]
[tex]\[ 289k^2 + 16k^2 = 1 \][/tex]
[tex]\[ 305k^2 = 1 \][/tex]
[tex]\[ k^2 = \frac{1}{305} \][/tex]
[tex]\[ k = \frac{1}{\sqrt{305}} \][/tex]

So,
[tex]\[ \sin \theta = 4k = \frac{4}{\sqrt{305}} \approx 0.229 \][/tex]
[tex]\[ \cos \theta = 17k = \frac{17}{\sqrt{305}} \approx 0.973 \][/tex]

3. Calculate the numerator: [tex]\((2 + 2 \sin \theta)(1 - \sin \theta)\)[/tex]:
[tex]\[ 2 + 2 \sin \theta = 2 + 2 \times 0.229 \approx 2.458 \][/tex]
[tex]\[ 1 - \sin \theta = 1 - 0.229 = 0.771 \][/tex]
[tex]\[ \text{Numerator:} (2 + 2 \sin \theta)(1 - \sin \theta) \approx 2.458 \times 0.771 \approx 1.895 \][/tex]

4. Calculate the denominator: [tex]\((1 + \cos \theta)(2 - 2 \cos \theta)\)[/tex]:
[tex]\[ 1 + \cos \theta = 1 + 0.973 \approx 1.973 \][/tex]
[tex]\[ 2 - 2 \cos \theta = 2 - 2 \times 0.973 \approx 2 - 1.946 = 0.054 \][/tex]
[tex]\[ \text{Denominator:} (1 + \cos \theta)(2 - 2 \cos \theta) \approx 1.973 \times 0.054 \approx 0.105 \][/tex]

5. Evaluate the expression:
[tex]\[ \frac{(2 + 2 \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(2 - 2 \cos \theta)} \approx \frac{1.895}{0.105} \approx 18.062 \][/tex]

Thus, the value of the expression is:
[tex]\[ \frac{(2+2 \sin \theta)(1-\sin \theta)}{(1+\cos \theta)(2-2 \cos \theta)} \approx 18.062 \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.