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Given that
sin A=(sqrt22)/(5) and
sin B=(sqrt7)/(3), and that angles
A and
B are both in Quadrant I, find the exact value of
cos(A-B), in simplest radical form.
Answer:

Given that $\sin A=\frac{\sqrt{22}}{5}$ and $\sin B=\frac{\sqrt{7}}{3}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\cos (A-B)$ , in simplest radical form. $\newline$ Answer:

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Find trigonometric ratios using multiple identities

Full solution

Q. Given that

\sin A=\frac{\sqrt{22}}{5}

and

\sin B=\frac{\sqrt{7}}{3}

, and that angles

A

and

B

are both in Quadrant I, find the exact value of

\cos (A-B)

, in simplest radical form.

\newline

Answer:

Apply cosine difference identity: Use the cosine difference identity: $\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)$ .
Find $\cos(A)$ and $\cos(B)$ : Find $\cos(A)$ and $\cos(B)$ using the Pythagorean identity: $\cos^2(A) = 1 - \sin^2(A)$ and $\cos^2(B) = 1 - \sin^2(B)$ .
Calculate $\cos^2(A)$ : Calculate $\cos^2(A)$ : $\cos^2(A) = 1 - \sin^2(A) = 1 - (\sqrt{22}/5)^2 = 1 - (22/25) = (25/25) - (22/25) = 3/25$ .
Calculate $\cos(A)$ : Calculate $\cos(A)$ : Since $A$ is in Quadrant I, $\cos(A)$ is positive. Therefore, $\cos(A) = \sqrt{\frac{3}{25}} = \frac{\sqrt{3}}{5}$ .
Calculate $\cos^2(B)$ : Calculate $\cos^2(B)$ : $\cos^2(B) = 1 - \sin^2(B) = 1 - (\sqrt{7}/3)^2 = 1 - (7/9) = (9/9) - (7/9) = 2/9$ .
Calculate $\cos(B)$ : Calculate $\cos(B)$ : Since $B$ is in Quadrant I, $\cos(B)$ is positive. Therefore, $\cos(B) = \sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3}$ .
Substitute values into identity: Substitute the values of $\cos(A)$ , $\cos(B)$ , $\sin(A)$ , and $\sin(B)$ into the cosine difference identity: $\cos(A-B) = \left(\frac{\sqrt{3}}{5}\right)\left(\frac{\sqrt{2}}{3}\right) + \left(\frac{\sqrt{22}}{5}\right)\left(\frac{\sqrt{7}}{3}\right)$ .
Simplify the expression: Simplify the expression: $\cos(A-B) = \frac{\sqrt{6}}{15} + \frac{\sqrt{154}}{15}$ .
Combine the terms: Combine the terms: $\cos(A-B) = (\sqrt{6} + \sqrt{154})/15$ .

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