If $\sin x=\frac{3}{5}, \cos y=-\frac{12}{13}$, where $x$ and $y$ both lie in second quadrant, find the value of $\sin (x+y)$.

Step 1: Find cos x and sin y

Since $x$ and $y$ both lie in the second quadrant, $\cos x$ will be negative and $\sin y$ will be positive. We use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find the missing trigonometric values.

Step 2: Calculate cos x

Substitute the given value of $\sin x = \frac{3}{5}$ into the formula for $\cos x$. Simplify the expression under the square root to find the value of $\cos x$.

Step 3: Calculate sin y

Substitute the given value of $\cos y = -\frac{12}{13}$ into the formula for $\sin y$. Simplify the expression under the square root to find the value of $\sin y$.

Step 4: Apply the sum formula for sine

Now that we have all the necessary trigonometric values, we can use the sum formula for sine to find $\sin(x+y)$.

Step 5: Substitute values and calculate

Substitute the calculated values of $\sin x$, $\cos y$, $\cos x$, and $\sin y$ into the sum formula. Perform the multiplication and then add the resulting fractions to get the final answer.

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