Given that cos A=(sqrt23)/(5) and sin B=(sqrt21)/(6), and that angles A and B are both in Quadrant I, find the exact value of sin(A+B), in simplest radical form. Answer:

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Given that  cos A=(sqrt23)/(5) and  sin B=(sqrt21)/(6), and that angles  A and  B are both in Quadrant I, find the exact value of  sin(A+B), in simplest radical form. Answer:

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Use Sine Addition Formula: Use the sine addition formula: sin(A+B) = sin(A)cos(B) + cos(A)sin(B). We need to find sin(A) and cos(B).Find sin(A) and cos(B): Since A is in Quadrant I, sin(A) is positive. We can find sin(A) using the Pythagorean identity: sin^2(A) + cos^2(A) = 1. We know cos(A) = (sqrt23)/5, so cos^2(A) = (23/25). sin^2(A) = 1 - cos^2(A) = 1 - (23/25) = (25/25) - (23/25) = (2/25). sin(A) = sqrt(sin^2(A)) = sqrt(2/25) = sqrt(2)/5.Find sin(B) and cos(B): Since B is in Quadrant I, cos(B) is also positive. We can find cos(B) using the Pythagorean identity: sin^2(B) + cos^2(B) = 1. We know sin(B) = (sqrt21)/6, so sin^2(B) = (21/36). cos^2(B) = 1 - sin^2(B) = 1 - (21/36) = (36/36) - (21/36) = (15/36) = (5/12) after simplifying. cos(B) = sqrt(cos^2(B)) = sqrt(5/12) = sqrt(5)/sqrt(12) = sqrt(5)/(2*sqrt(3)) = (sqrt(5)*sqrt(3))/(2*3) = (sqrt(15))/6 after rationalizing the denominator.Plug in Values: Now we have sin(A) = sqrt(2)/5 and cos(B) = (sqrt(15))/6. Plug these values into the sine addition formula: sin(A+B) = sin(A)cos(B) + cos(A)sin(B). sin(A+B) = (sqrt(2)/5)*(sqrt(15))/6 + (sqrt(23)/5)*(sqrt(21)/6).Simplify Expression: Simplify the expression: sin(A+B) = (sqrt(2)*sqrt(15))/(5*6) + (sqrt(23)*sqrt(21))/(5*6). sin(A+B) = (sqrt(30))/30 + (sqrt(483))/30.Combine Terms: Combine the terms over a common denominator: sin(A+B) = (sqrt(30) + sqrt(483))/30. Since sqrt(483) can be simplified to sqrt(3*161) = sqrt(3)*sqrt(161) = sqrt(3)*sqrt(7*23) = sqrt(3)*sqrt(7)*sqrt(23), we can rewrite the expression. sin(A+B) = (sqrt(30) + sqrt(3)*sqrt(7)*sqrt(23))/30.Simplify Further: Simplify the expression further: sin(A+B) = (sqrt(30) + sqrt(3)*sqrt(7)*sqrt(23))/30. sin(A+B) = (sqrt(30) + sqrt(3)*sqrt(7)*sqrt(23))/30 = (sqrt(30) + sqrt(3*7*23))/30. sin(A+B) = (sqrt(30) + sqrt(3*7*23))/30 = (sqrt(30) + sqrt(3*7*23))/30 = (sqrt(30) + sqrt(483))/30.Final Expression: Notice that sqrt(483) was already simplified in a previous step, so we do not need to simplify it again. The final expression for sin(A+B) is (sqrt(30) + sqrt(483))/30.

Given that $\cos A=\frac{\sqrt{23}}{5}$ and $\sin B=\frac{\sqrt{21}}{6}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\sin (A+B)$ , in simplest radical form. $\newline$ Answer:

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Given that cos⁡A=235 \cos A=\frac{\sqrt{23}}{5} cosA=523​​ and sin⁡B=216 \sin B=\frac{\sqrt{21}}{6} sinB=621​​, and that angles A A A and B B B are both in Quadrant I, find the exact value of sin⁡(A+B) \sin (A+B) sin(A+B), in simplest radical form.\newlineAnswer:

Given that $\cos A=\frac{\sqrt{23}}{5}$ and $\sin B=\frac{\sqrt{21}}{6}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\sin (A+B)$ , in simplest radical form. $\newline$ Answer: