We know that angles a and b are in the first quadrant. We also know this values:

We have to find sin(a+b).
We can use the following identity:

For the second term, we can replace the factors with another identity:
![\sin (a+b)=\sin a\cdot\cos b+\sqrt[]{1-\sin^2a}\cdot\sqrt[]{1-\cos^2b}](https://img.qammunity.org/2023/formulas/mathematics/college/letxqwvmpgts4v55e1nmg7g5lsu7oh224q.png)
Now we know all the terms from the right side of the equation and we can calculate:
![\begin{gathered} \sin (a+b)=\sin a\cdot\cos b+\sqrt[]{1-\sin^2a}\cdot\sqrt[]{1-\cos^2b} \\ \sin (a+b)=(5)/(13)\cdot(3)/(5)+\sqrt[]{1-((5)/(13))^2}\cdot\sqrt[]{1-((3)/(5))^2} \\ \sin (a+b)=(15)/(65)+\sqrt[]{1-(25)/(169)}\cdot\sqrt[]{1-(9)/(25)} \\ \sin (a+b)=(15)/(65)+\sqrt[]{(169-25)/(169)}\cdot\sqrt[]{(25-9)/(25)} \\ \sin (a+b)=(15)/(65)+\sqrt[]{(144)/(169)}\cdot\sqrt[]{(16)/(25)} \\ \sin (a+b)=(15)/(65)+(12)/(13)\cdot(4)/(5) \\ \sin (a+b)=(15)/(65)+(48)/(65) \\ \sin (a+b)=(63)/(65) \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/q4p6yd66pjryiecwkmrz5c9avog2qqgoic.png)
Answer: sin(a+b) = 63/65