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We are given:

\[
\tan(A^\circ) = \frac{3}{4} \quad \text{and} \quad \tan(B^\circ) = \frac{1}{2}
\]

We need to find \(\sin(A + B)^\circ\).

To solve this, we can use the trigonometric identity for the sine of a sum:

\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]

### Step 1: Finding \(\sin(A)\) and \(\cos(A)\)

From \(\tan(A) = \frac{3}{4}\), we know:

\[
\tan(A) = \frac{\sin(A)}{\cos(A)} = \frac{3}{4}
\]

This implies that we can consider a right triangle with opposite side = 3 and adjacent side = 4. Using the Pythagorean theorem, the hypotenuse would be:

\[
\text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Thus, we have:

\[
\sin(A) = \frac{3}{5}, \quad \cos(A) = \frac{4}{5}
\]

### Step 2: Finding \(\sin(B)\) and \(\cos(B)\)

Similarly, for \(\tan(B) = \frac{1}{2}\), we can imagine a right triangle with opposite side = 1 and adjacent side = 2. The hypotenuse is:

\[
\text{hypotenuse} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\]

Thus:

\[
\sin(B) = \frac{1}{\sqrt{5}}, \quad \cos(B) = \frac{2}{\sqrt{5}}
\]

### Step 3: Applying the Sum Formula

Now, we can substitute these values into the formula for \(\sin(A + B)\):

\[
\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
\]

Substituting the known values:

\[
\sin(A + B) = \left(\frac{3}{5} \times \frac{2}{\sqrt{5}}\right) + \left(\frac{4}{5} \times \frac{1}{\sqrt{5}}\right)
\]

Simplifying each term:

\[
\sin(A + B) = \frac{6}{5\sqrt{5}} + \frac{4}{5\sqrt{5}} = \frac{6 + 4}{5\sqrt{5}} = \frac{10}{5\sqrt{5}} = \frac{2}{\sqrt{5}}
\]

Rationalizing the denominator:

\[
\sin(A + B) = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
\]

Thus, the value of \(\sin(A + B)\) is:

\[
\sin(A + B) = \frac{2\sqrt{5}}{5}
\]

Would you like more details on any step? 

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Here are some related questions you might find interesting:
1. How do we find the exact values of other trigonometric functions like \(\cos(A + B)\)?
2. Can you explain how to derive the sine addition formula?
3. What is the importance of rationalizing denominators in trigonometric results?
4. How can we use the Pythagorean identity to check the accuracy of sine and cosine values?
5. What other trigonometric identities can help in solving angles sums or differences?

**Tip:** Always use the right triangle representation when working with tangent values to ensure accurate results for sine and cosine!

Given tanA° = 3/4 and tanB° = 1/2, find sin(A + B)°.

Learn how to solve for sin(A + B) using given values of tan(A) = 3/4 and tan(B) = 1/2. This step-by-step solution uses the sine addition formula and the Pythagorean Theorem to calculate exact values for sine and cosine functions. Suitable for high school students in Grades 10-12.

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