Given that $\sin x=\frac{1}{\sqrt{2}}$ and $\cos y=\frac{3}{\sqrt{10}}$ , 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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Find trigonometric ratios using multiple identities

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Q. Given that

\sin x=\frac{1}{\sqrt{2}}

and

\cos y=\frac{3}{\sqrt{10}}

, 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 Subtraction Formula: Use the sine subtraction formula: $\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)$ . We already know $\sin(x) = \frac{1}{\sqrt{2}}$ and $\cos(y) = \frac{3}{\sqrt{10}}$ . Next, we need to find $\cos(x)$ and $\sin(y)$ .
Find $\cos(x)$ and $\sin(y)$ : Since $x$ is in Quadrant I and $\sin(x) = \frac{1}{\sqrt{2}}$ , we can use the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to find $\cos(x)$ .
$\cos^2(x) = 1 - \sin^2(x)$
$\cos^2(x) = 1 - \left(\frac{1}{\sqrt{2}}\right)^2$
$\cos^2(x) = 1 - \frac{1}{2}$
$\cos^2(x) = \frac{1}{2}$
$\sin(y)$ $0$ , since $\cos(x)$ is positive in Quadrant I.
Substitute Values: Similarly, since $y$ is in Quadrant I and $\cos(y) = \frac{3}{\sqrt{10}}$ , we use the Pythagorean identity $\sin^2(y) + \cos^2(y) = 1$ to find $\sin(y)$ . $\sin^2(y) = 1 - \cos^2(y)$ $\sin^2(y) = 1 - \left(\frac{3}{\sqrt{10}}\right)^2$ $\sin^2(y) = 1 - \frac{9}{10}$ $\sin^2(y) = \frac{1}{10}$ $\sin(y) = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}}$ , since $\sin(y)$ is positive in Quadrant I.
Simplify Expression: Now we can substitute the values into the sine subtraction formula: $\newline$ $\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)$ $\newline$ $\sin(x - y) = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{3}{\sqrt{10}}\right) - \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{10}}\right)$
Rationalize Denominator: Simplify the expression: $\newline$ $\sin(x - y) = \frac{3}{\sqrt{2}\sqrt{10}} - \frac{1}{\sqrt{2}\sqrt{10}}$ $\newline$ $\sin(x - y) = \frac{3 - 1}{\sqrt{2}\sqrt{10}}$ $\newline$ $\sin(x - y) = \frac{2}{\sqrt{2}\sqrt{10}}$
Rationalize Denominator: Simplify the expression: $\newline$ $\sin(x - y) = \frac{3}{\sqrt{2}\sqrt{10}} - \frac{1}{\sqrt{2}\sqrt{10}}$ $\newline$ $\sin(x - y) = \frac{3 - 1}{\sqrt{2}\sqrt{10}}$ $\newline$ $\sin(x - y) = \frac{2}{\sqrt{2}\sqrt{10}}$ Rationalize the denominator: $\newline$ $\sin(x - y) = \frac{2}{\sqrt{2}\sqrt{10}} \times \frac{\sqrt{2}\sqrt{10}}{\sqrt{2}\sqrt{10}}$ $\newline$ $\sin(x - y) = \frac{2\sqrt{2}\sqrt{10}}{2\times10}$ $\newline$ $\sin(x - y) = \frac{\sqrt{2}\sqrt{10}}{10}$ $\newline$ $\sin(x - y) = \frac{\sqrt{20}}{10}$ $\newline$ $\sin(x - y) = \frac{\sqrt{4\times5}}{10}$ $\newline$ $\sin(x - y) = \frac{2\sqrt{5}}{10}$ $\newline$ $\sin(x - y) = \frac{\sqrt{5}}{5}$