Given that $\tan x=\sqrt{3}$ and $\cos y=\frac{\sqrt{2}}{2}$ , 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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Q. Given that

\tan x=\sqrt{3}

and

\cos y=\frac{\sqrt{2}}{2}

, 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:

Find $\sin x$ : Use the given information to find the values of $\sin x$ and $\sin y$ . Since $\tan x = \sqrt{3}$ , we know that $\frac{\sin x}{\cos x} = \sqrt{3}$ . We also know that $\cos^2 x + \sin^2 x = 1$ (Pythagorean identity). We need to find $\sin x$ .
Calculate $\cos x$ : Calculate $\cos x$ using the Pythagorean identity. $\newline$ Since $\tan x = \frac{\sin x}{\cos x} = \sqrt{3}$ , we can assume a right triangle where the opposite side is $\sqrt{3}$ and the adjacent side is $1$ (since tan is opposite over adjacent). Therefore, the hypotenuse is $\sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$ . So, $\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2}$ .
Calculate $\sin x$ : Calculate $\sin x$ using the Pythagorean identity. $\newline$ Now that we have $\cos x$ , we can use the identity $\sin^2 x + \cos^2 x = 1$ to find $\sin x$ . We have $\cos x = \frac{1}{2}$ , so $\cos^2 x = (\frac{1}{2})^2 = \frac{1}{4}$ . Therefore, $\sin^2 x = 1 - \frac{1}{4} = \frac{3}{4}$ , and $\sin x = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$ .
Find $\sin y$ : Use the given information to find $\sin y$ . We are given $\cos y = \frac{\sqrt{2}}{2}$ . Using the Pythagorean identity $\sin^2 y + \cos^2 y = 1$ , we can find $\sin y$ . We have $\cos^2 y = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}$ . Therefore, $\sin^2 y = 1 - \frac{1}{2} = \frac{1}{2}$ , and $\sin y = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}$ .
Use angle sum identity: Use the angle sum identity for cosine to find $\cos(x+y)$ . The angle sum identity for cosine is $\cos(x+y) = \cos x \cdot \cos y - \sin x \cdot \sin y$ . We have already found $\cos x$ , $\cos y$ , $\sin x$ , and $\sin y$ .
Substitute and simplify: Substitute the values into the angle sum identity and simplify. $\newline$ $\cos(x+y) = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ $\newline$ $\cos(x+y) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$ $\newline$ $\cos(x+y) = \frac{\sqrt{2} - \sqrt{6}}{4}$