Given that $\tan x=\frac{1}{\sqrt{3}}$ and $\cos y=\frac{\sqrt{5}}{5}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\cos (x+y)$ , in simplest radical form. $\newline$ Answer:

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Find trigonometric ratios using multiple identities

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Q. Given that

\tan x=\frac{1}{\sqrt{3}}

and

\cos y=\frac{\sqrt{5}}{5}

, and that angles

x

and

y

are both in Quadrant I, find the exact value of

\cos (x+y)

, in simplest radical form.

\newline

Answer:

Use Cosine Addition Formula: To find $\cos(x+y)$ , we can use the cosine addition formula: $\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)$ . We already have $\cos(y)$ , but we need to find $\cos(x)$ and $\sin(x)$ as well as $\sin(y)$ .
Find $\cos(x)$ and $\sin(x)$ : Since $\tan x = \frac{1}{\sqrt{3}}$ , we can create a right triangle where the opposite side is $1$ and the adjacent side is $\sqrt{3}$ . The hypotenuse, using the Pythagorean theorem, is $\sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$ . Therefore, $\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}$ .
Find $\sin(y)$ : Similarly, $\sin(x)$ can be found using the right triangle, where $\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}$ .
Substitute Values: To find $\sin(y)$ , we use the Pythagorean identity $\sin^2(y) + \cos^2(y) = 1$ . We know $\cos(y) = \sqrt{5}/5$ , so $\cos^2(y) = (\sqrt{5}/5)^2 = 5/25 = 1/5$ . Therefore, $\sin^2(y) = 1 - 1/5 = 4/5$ . Since $y$ is in Quadrant I, $\sin(y)$ is positive, so $\sin(y) = \sqrt{4/5} = 2/\sqrt{5} = 2\sqrt{5}/5$ after rationalizing the denominator.
Simplify Expression: Now we have all the necessary values: $\cos(x) = \sqrt{3}/2$ , $\sin(x) = 1/2$ , $\cos(y) = \sqrt{5}/5$ , and $\sin(y) = 2\sqrt{5}/5$ . We can substitute these into the cosine addition formula: $\cos(x+y) = (\sqrt{3}/2)(\sqrt{5}/5) - (1/2)(2\sqrt{5}/5)$ .
Final Answer: Simplify the expression: $\cos(x+y) = (\sqrt{3}\sqrt{5}/10) - (\sqrt{5}/5) = (\sqrt{15}/10) - (2\sqrt{5}/10) = (\sqrt{15} - 2\sqrt{5})/10$ .
Final Answer: Simplify the expression: $\cos(x+y) = (\sqrt{3}\sqrt{5}/10) - (\sqrt{5}/5) = (\sqrt{15}/10) - (2\sqrt{5}/10) = (\sqrt{15} - 2\sqrt{5})/10$ . The expression $(\sqrt{15} - 2\sqrt{5})/10$ is already in simplest radical form, so this is our final answer.