Get expert advice and community support on IDNLearn.com. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.

Given that [tex]\sin x = \frac{3}{5}[/tex] and [tex]\cos y = \frac{7}{25}[/tex], with [tex]\(x\)[/tex] and [tex]\(y\)[/tex] both being Quadrant I angles, find [tex]\cos (x+y)[/tex].

Sagot :

To solve for [tex]\(\cos(x + y)\)[/tex] given that [tex]\(\sin x = \frac{3}{5}\)[/tex] and [tex]\(\cos y = \frac{7}{25}\)[/tex] with [tex]\(x\)[/tex] and [tex]\(y\)[/tex] both being angles in the first quadrant, we will follow these steps:

1. Find [tex]\(\cos x\)[/tex]:
Since [tex]\(x\)[/tex] is in the first quadrant, both sine and cosine are positive. Using the Pythagorean identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex],
[tex]\[ \sin^2 x = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \][/tex]
[tex]\[ \cos^2 x = 1 - \sin^2 x = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \][/tex]
[tex]\[ \cos x = \sqrt{\frac{16}{25}} = \frac{4}{5} \][/tex]
Therefore, [tex]\(\cos x = \frac{4}{5}\)[/tex].

2. Find [tex]\(\sin y\)[/tex]:
Similarly, since [tex]\(y\)[/tex] is in the first quadrant, both sine and cosine are positive. Using the Pythagorean identity [tex]\(\cos^2 y + \sin^2 y = 1\)[/tex],
[tex]\[ \cos^2 y = \left(\frac{7}{25}\right)^2 = \frac{49}{625} \][/tex]
[tex]\[ \sin^2 y = 1 - \cos^2 y = 1 - \frac{49}{625} = \frac{625}{625} - \frac{49}{625} = \frac{576}{625} \][/tex]
[tex]\[ \sin y = \sqrt{\frac{576}{625}} = \frac{24}{25} \][/tex]
Therefore, [tex]\(\sin y = \frac{24}{25}\)[/tex].

3. Calculate [tex]\(\cos(x + y)\)[/tex]:
Using the angle addition formula for cosine: [tex]\(\cos(x + y) = \cos x \cos y - \sin x \sin y\)[/tex],
[tex]\[ \cos(x + y) = \left(\frac{4}{5} \cdot \frac{7}{25}\right) - \left(\frac{3}{5} \cdot \frac{24}{25}\right) \][/tex]
[tex]\[ \cos(x + y) = \frac{28}{125} - \frac{72}{125} = \frac{28 - 72}{125} = \frac{-44}{125} = -0.352 \][/tex]

Therefore, the values we have are:
[tex]\[ \cos x = \frac{4}{5} = 0.8 \][/tex]
[tex]\[ \sin y = \frac{24}{25} = 0.96 \][/tex]
[tex]\[ \cos(x + y) = -0.352 \][/tex]

So, when given [tex]\(\sin x = \frac{3}{5}\)[/tex] and [tex]\(\cos y = \frac{7}{25}\)[/tex] with angles in the first quadrant, the value of [tex]\(\cos(x + y)\)[/tex] is [tex]\(-0.352\)[/tex].
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.