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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]
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]
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