Question

For positive acute angles A and B, it is known that sin A = 3/5; and cos B = 8/17.Find the value of sin(A + B) in simplest form.

Answer 1

1) Given the values of the positive acute angles A and B

2) Let's calculate sin(A+B), let's then find the cos(A)

`\begin{gathered} \sin ^2(A)\text{ +}\cos ^2(A)=1 \\ \cos ^2(A)=1-((3)/(5))^2 \\ \cos ^{}(A)=\sqrt[]{1-(9)/(25)} \\ \cos (A)=(4)/(5) \end{gathered}`

Now let's find the sin (B), using the same Pythagorean Identity

`\begin{gathered} \sin ^2(B)=1-\cos ^2(B) \\ \sin (B)\text{ =}\sqrt[]{1-((8)/(17))^2} \\ \sin (B)\text{ =}(15)/(17) \end{gathered}`

3)Finally, let's calculate the sin (A+B)

`\begin{gathered} \sin (A+B)\text{ =}\sin (A)\cos B+\cos A\sin B \\ \sin (A+B)=(3)/(5)*(8)/(17)+(4)/(5)*(15)/(17)\text{ =}(84)/(85) \end{gathered}`

sin(A+B) = 84/85