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Sagot :
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]
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]
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