Given that sine, A, equals, start fraction, 3, divided by, 4, end fractionsinA= 
4
3
​	
  and sine, B, equals, start fraction, square root of, 8, end square root, divided by, 5, end fractionsinB= 
5
8
​	
 
​	
 , and that angles AA and BB are both in Quadrant I, find the exact value of sine, left parenthesis, A, plus, B, right parenthesissin(A+B), in simplest radical form.


We are given that:

- \(\sin(A) = \frac{3}{4}\) 
- \(\sin(B) = \frac{\sqrt{8}}{5}\)

and both angles \(A\) and \(B\) are in Quadrant I. We need to find the exact value of \(\sin(A + B)\).

### Step 1: Recall the sum of angles formula for sine:
\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]

### Step 2: Find \(\cos(A)\) and \(\cos(B)\) using the Pythagorean identity:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

For \(A\):
\[
\sin(A) = \frac{3}{4}, \quad \cos^2(A) = 1 - \sin^2(A) = 1 - \left(\frac{3}{4}\right)^2 = 1 - \frac{9}{16} = \frac{7}{16}
\]
\[
\cos(A) = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}
\]

For \(B\):
\[
\sin(B) = \frac{\sqrt{8}}{5}, \quad \cos^2(B) = 1 - \sin^2(B) = 1 - \left(\frac{\sqrt{8}}{5}\right)^2 = 1 - \frac{8}{25} = \frac{17}{25}
\]
\[
\cos(B) = \sqrt{\frac{17}{25}} = \frac{\sqrt{17}}{5}
\]

### Step 3: Plug values into the sum formula:
\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]
\[
\sin(A + B) = \left(\frac{3}{4}\right)\left(\frac{\sqrt{17}}{5}\right) + \left(\frac{\sqrt{7}}{4}\right)\left(\frac{\sqrt{8}}{5}\right)
\]
\[
\sin(A + B) = \frac{3\sqrt{17}}{20} + \frac{\sqrt{7}\sqrt{8}}{20}
\]
\[
\sin(A + B) = \frac{3\sqrt{17}}{20} + \frac{\sqrt{56}}{20}
\]
\[
\sin(A + B) = \frac{3\sqrt{17} + \sqrt{56}}{20}
\]

### Final Answer:
\[
\sin(A + B) = \frac{3\sqrt{17} + \sqrt{56}}{20}
\]

This is the exact value of \(\sin(A + B)\) in simplest radical form.

Do you want further details or have any questions? Here are some related questions:

1. How can we derive the sum of angles formula for sine?
2. What is the geometrical interpretation of sine and cosine in different quadrants?
3. How does the Pythagorean identity relate sine and cosine values?
4. What happens when angles \(A\) and \(B\) are in different quadrants?
5. How can we simplify expressions involving square roots?

**Tip:** When dealing with trigonometric identities, it's helpful to remember the quadrant where the angle lies, as it affects the signs of sine and cosine.

Given that sin(A) = 3/4 and sin(B) = sqrt(8)/5, and that angles A and B are both in Quadrant I, find the exact value of sin(A + B) in simplest radical form.

Find the exact value of sin(A + B) where sin(A) = 3/4 and sin(B) = sqrt(8)/5, using the sum of angles formula and the Pythagorean identity. Detailed solution suitable for students in Grades 10-12.

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