Answered

Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Assume that [tex]$0 \ \textless \ x \ \textless \ \frac{\pi}{2}$[/tex] and [tex]$0 \ \textless \ y \ \textless \ \frac{\pi}{2}$[/tex]. Find the exact value of [tex]\tan (x - y)[/tex] if [tex]\sin x = \frac{8}{17}[/tex] and [tex]\cos y = \frac{3}{5}[/tex].

A. [tex]\frac{36}{77}[/tex]
B. [tex]-\frac{36}{77}[/tex]
C. [tex]\frac{77}{36}[/tex]
D. [tex]-\frac{77}{36}[/tex]

Please select the best answer from the choices provided:

A
B
C
D

Sagot :

GinnyAnswer
We are given that [tex]\(\sin x = \frac{8}{17}\)[/tex] and [tex]\(\cos y = \frac{3}{5}\)[/tex]. We need to find [tex]\(\tan(x - y)\)[/tex].

First, we use the Pythagorean identity to find [tex]\(\cos x\)[/tex] and [tex]\(\sin y\)[/tex].

For [tex]\(\cos x\)[/tex]:
[tex]\[ \cos^2 x = 1 - \sin^2 x \][/tex]
[tex]\[ \cos^2 x = 1 - \left(\frac{8}{17}\right)^2 \][/tex]
[tex]\[ \cos^2 x = 1 - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 x = \frac{289}{289} - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 x = \frac{225}{289} \][/tex]
[tex]\[ \cos x = \sqrt{\frac{225}{289}} \][/tex]
[tex]\[ \cos x = \frac{15}{17} \][/tex]

For [tex]\(\sin y\)[/tex]:
[tex]\[ \sin^2 y = 1 - \cos^2 y \][/tex]
[tex]\[ \sin^2 y = 1 - \left(\frac{3}{5}\right)^2 \][/tex]
[tex]\[ \sin^2 y = 1 - \frac{9}{25} \][/tex]
[tex]\[ \sin^2 y = \frac{25}{25} - \frac{9}{25} \][/tex]
[tex]\[ \sin^2 y = \frac{16}{25} \][/tex]
[tex]\[ \sin y = \sqrt{\frac{16}{25}} \][/tex]
[tex]\[ \sin y = \frac{4}{5} \][/tex]

Next, we use the values of [tex]\(\sin x\)[/tex], [tex]\(\cos x\)[/tex], [tex]\(\sin y\)[/tex], and [tex]\(\cos y\)[/tex] to find [tex]\(\tan x\)[/tex] and [tex]\(\tan y\)[/tex].

[tex]\[ \tan x = \frac{\sin x}{\cos x} \][/tex]
[tex]\[ \tan x = \frac{\frac{8}{17}}{\frac{15}{17}} = \frac{8}{15} \][/tex]

[tex]\[ \tan y = \frac{\sin y}{\cos y} \][/tex]
[tex]\[ \tan y = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \][/tex]

Using the tangent subtraction formula:
[tex]\[ \tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \][/tex]
[tex]\[ \tan(x - y) = \frac{\frac{8}{15} - \frac{4}{3}}{1 + \frac{8}{15} \cdot \frac{4}{3}} \][/tex]
[tex]\[ \tan(x - y) = \frac{\frac{8}{15} - \frac{20}{15}}{1 + \frac{32}{45}} \][/tex]
[tex]\[ \tan(x - y) = \frac{\frac{8 - 20}{15}}{1 + \frac{32}{45}} \][/tex]
[tex]\[ \tan(x - y) = \frac{-\frac{12}{15}}{1 + \frac{32}{45}} \][/tex]
[tex]\[ \tan(x - y) = \frac{-\frac{4}{5}}{\frac{45 + 32}{45}} \][/tex]
[tex]\[ \tan(x - y) = \frac{-\frac{4}{5}}{\frac{77}{45}} \][/tex]
[tex]\[ \tan(x - y) = -\frac{4}{5} \cdot \frac{45}{77} \][/tex]
[tex]\[ \tan(x - y) = -\frac{180}{385} \][/tex]
[tex]\[ \tan(x - y) = -\frac{36}{77} \][/tex]

So, the exact value of [tex]\(\tan(x - y)\)[/tex] is [tex]\(-\frac{36}{77}\)[/tex], which corresponds to answer choice:

b. [tex]\(-\frac{36}{77}\)[/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.