Question $\newline$ Watch Video $\newline$ Show Examples $\newline$ Given that $\cos A=\frac{\sqrt{2}}{3}$ and $\sin B=\frac{\sqrt{3}}{4}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\sin (A-B)$ , in simplest radical form.

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Algebra 2

Find trigonometric ratios using multiple identities

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Q. Question

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Watch Video

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Show Examples

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Given that

\cos A=\frac{\sqrt{2}}{3}

and

\sin B=\frac{\sqrt{3}}{4}

, and that angles

A

and

B

are both in Quadrant I, find the exact value of

\sin (A-B)

, in simplest radical form.

Apply Sine Subtraction Formula: Use the sine subtraction formula: $\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B)$ .
Find $\sin(A)$ using Pythagorean Identity: Find $\sin(A)$ using the Pythagorean identity: $\sin^2(A) + \cos^2(A) = 1$ . $\newline$ $\sin^2(A) = 1 - \cos^2(A)$ $\newline$ $\sin^2(A) = 1 - (\frac{\sqrt{2}}{3})^2$ $\newline$ $\sin^2(A) = 1 - \frac{2}{9}$ $\newline$ $\sin^2(A) = \frac{9}{9} - \frac{2}{9}$ $\newline$ $\sin^2(A) = \frac{7}{9}$ $\newline$ $\sin(A) = \sqrt{\frac{7}{9}}$ $\newline$ $\sin(A) = \frac{\sqrt{7}}{3}$ .
Find $\cos(B)$ using Pythagorean Identity: Find $\cos(B)$ using the Pythagorean identity: $\sin^2(B) + \cos^2(B) = 1.$ $\newline$ $\cos^2(B) = 1 - \sin^2(B)$ $\newline$ $\cos^2(B) = 1 - (\frac{\sqrt{3}}{4})^2$ $\newline$ $\cos^2(B) = 1 - \frac{3}{16}$ $\newline$ $\cos^2(B) = \frac{16}{16} - \frac{3}{16}$ $\newline$ $\cos^2(B) = \frac{13}{16}$ $\newline$ $\cos(B) = \sqrt{\frac{13}{16}}$ $\newline$ $\cos(B) = \frac{\sqrt{13}}{4}.$
Plug Values into Formula: Plug values into the sine subtraction formula: $\newline$ $\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B)$ $\newline$ $\sin(A-B) = \left(\frac{\sqrt{7}}{3}\right)\left(\frac{\sqrt{13}}{4}\right) - \left(\frac{\sqrt{2}}{3}\right)\left(\frac{\sqrt{3}}{4}\right)$ $\newline$ $\sin(A-B) = \frac{\sqrt{7\cdot13}}{12} - \frac{\sqrt{2\cdot3}}{12}$ $\newline$ $\sin(A-B) = \frac{\sqrt{91}}{12} - \frac{\sqrt{6}}{12}$ .
Combine Terms: Combine the terms: $\sin(A-B) = \frac{\sqrt{91} - \sqrt{6}}{12}$ .