Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Certainly! Let's go through the steps to prove that [tex]\(\cos 2\theta = \sin 4\phi\)[/tex] given [tex]\(\tan \theta = \frac{1}{7}\)[/tex] and [tex]\(\tan \phi = \frac{1}{3}\)[/tex].
### Step-by-Step Solution:
1. Represent [tex]\(\theta\)[/tex] and [tex]\(\phi\)[/tex] using arctangent:
Since [tex]\(\tan \theta = \frac{1}{7}\)[/tex], we have:
[tex]\[ \theta = \tan^{-1}\left(\frac{1}{7}\right) \][/tex]
Similarly, since [tex]\(\tan \phi = \frac{1}{3}\)[/tex], we have:
[tex]\[ \phi = \tan^{-1}\left(\frac{1}{3}\right) \][/tex]
2. Double-angle formula for cosine:
For [tex]\(\cos 2\theta\)[/tex], we use the double-angle identity for cosine in terms of tangent:
[tex]\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \][/tex]
Substituting [tex]\(\tan \theta = \frac{1}{7}\)[/tex]:
[tex]\[ \cos 2\theta = \frac{1 - \left(\frac{1}{7}\right)^2}{1 + \left(\frac{1}{7}\right)^2} = \frac{1 - \frac{1}{49}}{1 + \frac{1}{49}} = \frac{\frac{48}{49}}{\frac{50}{49}} = \frac{48}{50} = \frac{24}{25} \][/tex]
Thus,
[tex]\[ \cos 2\theta = 0.96 \][/tex]
3. Double-angle formula for sine:
To find [tex]\(\sin 4\phi\)[/tex], use the identity:
[tex]\[ \sin 4\phi = 2 \sin 2\phi \cos 2\phi \][/tex]
First, find [tex]\(\sin 2\phi\)[/tex] and [tex]\(\cos 2\phi\)[/tex] using the double-angle formulas for sine and cosine:
[tex]\[ \sin 2\phi = \frac{2 \tan \phi}{1 + \tan^2 \phi} \][/tex]
Substituting [tex]\(\tan \phi = \frac{1}{3}\)[/tex]:
[tex]\[ \sin 2\phi = \frac{2 \left(\frac{1}{3}\right)}{1 + \left(\frac{1}{3}\right)^2} = \frac{\frac{2}{3}}{1 + \frac{1}{9}} = \frac{\frac{2}{3}}{\frac{10}{9}} = \frac{2}{3} \times \frac{9}{10} = \frac{6}{10} = \frac{3}{5} \][/tex]
[tex]\[ \cos 2\phi = \frac{1 - \tan^2 \phi}{1 + \tan^2 \phi} = \frac{1 - \left(\frac{1}{3}\right)^2}{1 + \left(\frac{1}{3}\right)^2} = \frac{1 - \frac{1}{9}}{1 + \frac{1}{9}} = \frac{\frac{8}{9}}{\frac{10}{9}} = \frac{8}{10} = \frac{4}{5} \][/tex]
Now, calculate [tex]\(\sin 4\phi\)[/tex]:
[tex]\[ \sin 4\phi = 2 \sin 2\phi \cos 2\phi = 2 \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) = 2 \times \frac{12}{25} = \frac{24}{25} \][/tex]
Thus,
[tex]\[ \sin 4\phi = 0.96 \][/tex]
4. Conclusion:
We have found that:
[tex]\[ \cos 2\theta = 0.96 \quad \text{and} \quad \sin 4\phi = 0.96 \][/tex]
Therefore,
[tex]\[ \cos 2\theta = \sin 4\phi \][/tex]
Hence, it is proven that [tex]\(\cos 2\theta = \sin 4\phi\)[/tex].
### Step-by-Step Solution:
1. Represent [tex]\(\theta\)[/tex] and [tex]\(\phi\)[/tex] using arctangent:
Since [tex]\(\tan \theta = \frac{1}{7}\)[/tex], we have:
[tex]\[ \theta = \tan^{-1}\left(\frac{1}{7}\right) \][/tex]
Similarly, since [tex]\(\tan \phi = \frac{1}{3}\)[/tex], we have:
[tex]\[ \phi = \tan^{-1}\left(\frac{1}{3}\right) \][/tex]
2. Double-angle formula for cosine:
For [tex]\(\cos 2\theta\)[/tex], we use the double-angle identity for cosine in terms of tangent:
[tex]\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \][/tex]
Substituting [tex]\(\tan \theta = \frac{1}{7}\)[/tex]:
[tex]\[ \cos 2\theta = \frac{1 - \left(\frac{1}{7}\right)^2}{1 + \left(\frac{1}{7}\right)^2} = \frac{1 - \frac{1}{49}}{1 + \frac{1}{49}} = \frac{\frac{48}{49}}{\frac{50}{49}} = \frac{48}{50} = \frac{24}{25} \][/tex]
Thus,
[tex]\[ \cos 2\theta = 0.96 \][/tex]
3. Double-angle formula for sine:
To find [tex]\(\sin 4\phi\)[/tex], use the identity:
[tex]\[ \sin 4\phi = 2 \sin 2\phi \cos 2\phi \][/tex]
First, find [tex]\(\sin 2\phi\)[/tex] and [tex]\(\cos 2\phi\)[/tex] using the double-angle formulas for sine and cosine:
[tex]\[ \sin 2\phi = \frac{2 \tan \phi}{1 + \tan^2 \phi} \][/tex]
Substituting [tex]\(\tan \phi = \frac{1}{3}\)[/tex]:
[tex]\[ \sin 2\phi = \frac{2 \left(\frac{1}{3}\right)}{1 + \left(\frac{1}{3}\right)^2} = \frac{\frac{2}{3}}{1 + \frac{1}{9}} = \frac{\frac{2}{3}}{\frac{10}{9}} = \frac{2}{3} \times \frac{9}{10} = \frac{6}{10} = \frac{3}{5} \][/tex]
[tex]\[ \cos 2\phi = \frac{1 - \tan^2 \phi}{1 + \tan^2 \phi} = \frac{1 - \left(\frac{1}{3}\right)^2}{1 + \left(\frac{1}{3}\right)^2} = \frac{1 - \frac{1}{9}}{1 + \frac{1}{9}} = \frac{\frac{8}{9}}{\frac{10}{9}} = \frac{8}{10} = \frac{4}{5} \][/tex]
Now, calculate [tex]\(\sin 4\phi\)[/tex]:
[tex]\[ \sin 4\phi = 2 \sin 2\phi \cos 2\phi = 2 \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) = 2 \times \frac{12}{25} = \frac{24}{25} \][/tex]
Thus,
[tex]\[ \sin 4\phi = 0.96 \][/tex]
4. Conclusion:
We have found that:
[tex]\[ \cos 2\theta = 0.96 \quad \text{and} \quad \sin 4\phi = 0.96 \][/tex]
Therefore,
[tex]\[ \cos 2\theta = \sin 4\phi \][/tex]
Hence, it is proven that [tex]\(\cos 2\theta = \sin 4\phi\)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
