Given that $\cos x=\frac{4}{5}$ and $\sin y=\frac{\sqrt{8}}{3}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\sin (x+y)$ , in simplest radical form. $\newline$ Answer:

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Q. Given that

\cos x=\frac{4}{5}

and

\sin y=\frac{\sqrt{8}}{3}

, and that angles

x

and

y

are both in Quadrant I, find the exact value of

\sin (x+y)

, in simplest radical form.

\newline

Answer:

Apply Sine Addition Formula: Use the sine addition formula: $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ . First, we need to find $\sin(x)$ and $\cos(y)$ . Since $\cos x = \frac{4}{5}$ and $x$ is in Quadrant I, we can find $\sin(x)$ using the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ .
Calculate $\sin(x)$ : Calculate $\sin(x)$ :
$\sin^2(x) = 1 - \cos^2(x)$
$\sin^2(x) = 1 - \left(\frac{4}{5}\right)^2$
$\sin^2(x) = 1 - \frac{16}{25}$
$\sin^2(x) = \frac{25}{25} - \frac{16}{25}$
$\sin^2(x) = \frac{9}{25}$
$\sin(x) = \sqrt{\frac{9}{25}}$
$\sin(x) = \frac{3}{5}$
Since $x$ is in Quadrant I, $\sin(x)$ is positive.
Find $\cos(y)$ : Now, find $\cos(y)$ : We are given $\sin y = \sqrt{8}/3$ . Using the Pythagorean identity again, $\cos^2(y) + \sin^2(y) = 1$ . $\cos^2(y) = 1 - \sin^2(y)$ $\cos^2(y) = 1 - (\sqrt{8}/3)^2$ $\cos^2(y) = 1 - 8/9$ $\cos^2(y) = (9/9) - (8/9)$ $\cos^2(y) = 1/9$ $\cos(y) = \sqrt{1/9}$ $\cos(y)$ $0$ Since $\cos(y)$ $1$ is in Quadrant I, $\cos(y)$ is positive.
Use Sine Addition Formula: Now that we have $\sin(x) = \frac{3}{5}$ and $\cos(y) = \frac{1}{3}$ , we can use the sine addition formula: $\newline$ $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ $\newline$ $\sin(x+y) = \left(\frac{3}{5}\right)\left(\frac{1}{3}\right) + \left(\frac{4}{5}\right)\left(\frac{\sqrt{8}}{3}\right)$ $\newline$ $\sin(x+y) = \frac{3}{15} + \frac{4\sqrt{8}}{15}$ $\newline$ $\sin(x+y) = \frac{1}{5} + \frac{4\sqrt{8}}{15}$
Combine Terms: Combine the terms to get the final answer: $\newline$ $\sin(x+y) = \frac{1}{5}\left(\frac{3}{3}\right) + \frac{4\sqrt{8}}{15}$ $\newline$ $\sin(x+y) = \frac{3}{15} + \frac{4\sqrt{8}}{15}$ $\newline$ $\sin(x+y) = \frac{3 + 4\sqrt{8}}{15}$