From personal advice to professional guidance, IDNLearn.com has the answers you seek. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.

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 appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.