Get personalized answers to your specific questions with IDNLearn.com. Ask anything and receive prompt, well-informed answers from our community of experienced experts.
Sagot :
Sure, let's solve this step-by-step.
Given:
[tex]\[ \tan \theta=\frac{1}{7} \][/tex]
[tex]\[ \tan \beta=\frac{1}{3} \][/tex]
To find [tex]\(\cos 2\theta\)[/tex] and [tex]\(\sin 4\beta\)[/tex], we will use trigonometric identities.
1. Finding [tex]\(\cos 2\theta\)[/tex]:
The double-angle identity for cosine is:
[tex]\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \][/tex]
Substituting [tex]\(\tan \theta = \frac{1}{7}\)[/tex]:
[tex]\[ \cos 2\theta = \frac{1 - \left(\frac{1}{7}\right)^2}{1 + \left(\frac{1}{7}\right)^2} \][/tex]
[tex]\[ \cos 2\theta = \frac{1 - \frac{1}{49}}{1 + \frac{1}{49}} \][/tex]
[tex]\[ \cos 2\theta = \frac{\frac{49}{49} - \frac{1}{49}}{\frac{49}{49} + \frac{1}{49}} \][/tex]
[tex]\[ \cos 2\theta = \frac{\frac{48}{49}}{\frac{50}{49}} \][/tex]
[tex]\[ \cos 2\theta = \frac{48}{50} \][/tex]
[tex]\[ \cos 2\theta = 0.96 \][/tex]
2. Finding [tex]\(\sin 4\beta\)[/tex]:
The multiple-angle identity for sine is:
[tex]\[ \sin 4\beta = 2 \sin 2\beta \cos 2\beta \][/tex]
First, we need [tex]\(\sin 2\beta\)[/tex] and [tex]\(\cos 2\beta\)[/tex]. The double-angle formulas for sine and cosine are:
[tex]\[ \sin 2\beta = \frac{2 \tan \beta}{1 + \tan^2 \beta} \][/tex]
[tex]\[ \cos 2\beta = \frac{1 - \tan^2 \beta}{1 + \tan^2 \beta} \][/tex]
Substituting [tex]\(\tan \beta = \frac{1}{3} \)[/tex]:
[tex]\[ \sin 2\beta = \frac{2 \left(\frac{1}{3}\right)}{1 + \left(\frac{1}{3}\right)^2} \][/tex]
[tex]\[ \sin 2\beta = \frac{\frac{2}{3}}{1 + \frac{1}{9}} \][/tex]
[tex]\[ \sin 2\beta = \frac{\frac{2}{3}}{\frac{10}{9}} \][/tex]
[tex]\[ \sin 2\beta = \frac{2}{3} \cdot \frac{9}{10} \][/tex]
[tex]\[ \sin 2\beta = \frac{18}{30} \][/tex]
[tex]\[ \sin 2\beta = \frac{3}{5} \][/tex]
[tex]\[ \cos 2\beta = \frac{1 - \left(\frac{1}{3}\right)^2}{1 + \left(\frac{1}{3}\right)^2} \][/tex]
[tex]\[ \cos 2\beta = \frac{1 - \frac{1}{9}}{1 + \frac{1}{9}} \][/tex]
[tex]\[ \cos 2\beta = \frac{\frac{9}{9} - \frac{1}{9}}{\frac{9}{9} + \frac{1}{9}} \][/tex]
[tex]\[ \cos 2\beta = \frac{\frac{8}{9}}{\frac{10}{9}} \][/tex]
[tex]\[ \cos 2\beta = \frac{8}{10} \][/tex]
[tex]\[ \cos 2\beta = 0.8 \][/tex]
Now, using these results in the equation for [tex]\(\sin 4\beta\)[/tex]:
[tex]\[ \sin 4\beta = 2 \left(\frac{3}{5}\right) \left(0.8\right) \][/tex]
[tex]\[ \sin 4\beta = 2 \left(\frac{3 \times 8}{5 \times 10}\right) \][/tex]
[tex]\[ \sin 4\beta = 2 \left(\frac{24}{50}\right) \][/tex]
[tex]\[ \sin 4\beta = \frac{48}{50} \][/tex]
[tex]\[ \sin 4\beta = 0.96 \][/tex]
Thus, we have shown that:
[tex]\[ \cos 2\theta = \sin 4\beta \][/tex]
[tex]\[ 0.96 = 0.96 \][/tex]
Hence, [tex]\(\boxed{\cos 2 \theta = \sin 4 \beta}\)[/tex].
Given:
[tex]\[ \tan \theta=\frac{1}{7} \][/tex]
[tex]\[ \tan \beta=\frac{1}{3} \][/tex]
To find [tex]\(\cos 2\theta\)[/tex] and [tex]\(\sin 4\beta\)[/tex], we will use trigonometric identities.
1. Finding [tex]\(\cos 2\theta\)[/tex]:
The double-angle identity for cosine is:
[tex]\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \][/tex]
Substituting [tex]\(\tan \theta = \frac{1}{7}\)[/tex]:
[tex]\[ \cos 2\theta = \frac{1 - \left(\frac{1}{7}\right)^2}{1 + \left(\frac{1}{7}\right)^2} \][/tex]
[tex]\[ \cos 2\theta = \frac{1 - \frac{1}{49}}{1 + \frac{1}{49}} \][/tex]
[tex]\[ \cos 2\theta = \frac{\frac{49}{49} - \frac{1}{49}}{\frac{49}{49} + \frac{1}{49}} \][/tex]
[tex]\[ \cos 2\theta = \frac{\frac{48}{49}}{\frac{50}{49}} \][/tex]
[tex]\[ \cos 2\theta = \frac{48}{50} \][/tex]
[tex]\[ \cos 2\theta = 0.96 \][/tex]
2. Finding [tex]\(\sin 4\beta\)[/tex]:
The multiple-angle identity for sine is:
[tex]\[ \sin 4\beta = 2 \sin 2\beta \cos 2\beta \][/tex]
First, we need [tex]\(\sin 2\beta\)[/tex] and [tex]\(\cos 2\beta\)[/tex]. The double-angle formulas for sine and cosine are:
[tex]\[ \sin 2\beta = \frac{2 \tan \beta}{1 + \tan^2 \beta} \][/tex]
[tex]\[ \cos 2\beta = \frac{1 - \tan^2 \beta}{1 + \tan^2 \beta} \][/tex]
Substituting [tex]\(\tan \beta = \frac{1}{3} \)[/tex]:
[tex]\[ \sin 2\beta = \frac{2 \left(\frac{1}{3}\right)}{1 + \left(\frac{1}{3}\right)^2} \][/tex]
[tex]\[ \sin 2\beta = \frac{\frac{2}{3}}{1 + \frac{1}{9}} \][/tex]
[tex]\[ \sin 2\beta = \frac{\frac{2}{3}}{\frac{10}{9}} \][/tex]
[tex]\[ \sin 2\beta = \frac{2}{3} \cdot \frac{9}{10} \][/tex]
[tex]\[ \sin 2\beta = \frac{18}{30} \][/tex]
[tex]\[ \sin 2\beta = \frac{3}{5} \][/tex]
[tex]\[ \cos 2\beta = \frac{1 - \left(\frac{1}{3}\right)^2}{1 + \left(\frac{1}{3}\right)^2} \][/tex]
[tex]\[ \cos 2\beta = \frac{1 - \frac{1}{9}}{1 + \frac{1}{9}} \][/tex]
[tex]\[ \cos 2\beta = \frac{\frac{9}{9} - \frac{1}{9}}{\frac{9}{9} + \frac{1}{9}} \][/tex]
[tex]\[ \cos 2\beta = \frac{\frac{8}{9}}{\frac{10}{9}} \][/tex]
[tex]\[ \cos 2\beta = \frac{8}{10} \][/tex]
[tex]\[ \cos 2\beta = 0.8 \][/tex]
Now, using these results in the equation for [tex]\(\sin 4\beta\)[/tex]:
[tex]\[ \sin 4\beta = 2 \left(\frac{3}{5}\right) \left(0.8\right) \][/tex]
[tex]\[ \sin 4\beta = 2 \left(\frac{3 \times 8}{5 \times 10}\right) \][/tex]
[tex]\[ \sin 4\beta = 2 \left(\frac{24}{50}\right) \][/tex]
[tex]\[ \sin 4\beta = \frac{48}{50} \][/tex]
[tex]\[ \sin 4\beta = 0.96 \][/tex]
Thus, we have shown that:
[tex]\[ \cos 2\theta = \sin 4\beta \][/tex]
[tex]\[ 0.96 = 0.96 \][/tex]
Hence, [tex]\(\boxed{\cos 2 \theta = \sin 4 \beta}\)[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.
