To find the exact value of the expression

$\sin\left(\frac{\pi}{4}\right)\cos\left(\frac{5\pi}{12}\right) + \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{5\pi}{12}\right),$

we can recognize this as a form of the sine addition formula:

$\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B).$

In this case:

• $A = \frac{\pi}{4}$
• $B = \frac{5\pi}{12}$

Thus, the expression simplifies to:

$\sin\left(\frac{\pi}{4} + \frac{5\pi}{12}\right).$

Step 1: Simplify the Angle

First, add the angles:

$\frac{\pi}{4} + \frac{5\pi}{12}.$

To add these, we need a common denominator. The least common multiple of 4 and 12 is 12, so rewrite $\frac{\pi}{4}$ as $\frac{3\pi}{12}$:

$\frac{3\pi}{12} + \frac{5\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3}.$

Step 2: Find $\sin\left(\frac{2\pi}{3}\right)$

Now, we evaluate $\sin\left(\frac{2\pi}{3}\right)$. The angle $\frac{2\pi}{3}$ is in the second quadrant, where sine is positive. It is equivalent to $\pi - \frac{\pi}{3}$, so:

$\sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right).$

Since $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$, we conclude:

$\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}.$

Final Answer

The exact value of the expression is:

$\frac{\sqrt{3}}{2}.$

Would you like a further breakdown of any steps, or do you have any other questions?

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Here are five related questions you might find helpful:

1. How do we derive the sine addition formula?
2. What are some applications of the cosine addition formula?
3. How can we determine the quadrant of an angle and apply trigonometric identities accordingly?
4. Why is $\sin\left(\pi - \theta\right) = \sin(\theta)$ important for solving trigonometric expressions?
5. How do we find the exact values of trigonometric functions for common angles?

Tip: Remember that recognizing common trigonometric identities can simplify complex expressions quickly.