https://file.aiquickdraw.com/mathbot/file/1727033747026il2yvbm5.jpg

Let's solve the two trigonometric expressions from the image.

### Part (a)
\[
\sin\left(\frac{5\pi}{24}\right) \cos\left(\frac{\pi}{24}\right)
\]

This expression can be evaluated using a product-to-sum trigonometric identity:

\[
\sin(A)\cos(B) = \frac{1}{2} \left[\sin(A + B) + \sin(A - B)\right]
\]

Substituting \(A = \frac{5\pi}{24}\) and \(B = \frac{\pi}{24}\):

\[
\sin\left(\frac{5\pi}{24}\right)\cos\left(\frac{\pi}{24}\right) = \frac{1}{2} \left[\sin\left(\frac{5\pi}{24} + \frac{\pi}{24}\right) + \sin\left(\frac{5\pi}{24} - \frac{\pi}{24}\right)\right]
\]
\[
= \frac{1}{2} \left[\sin\left(\frac{6\pi}{24}\right) + \sin\left(\frac{4\pi}{24}\right)\right]
\]
\[
= \frac{1}{2} \left[\sin\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{6}\right)\right]
\]

Now use the known values for \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\):

\[
= \frac{1}{2} \left[\frac{\sqrt{2}}{2} + \frac{1}{2}\right]
\]
\[
= \frac{1}{2} \times \frac{\sqrt{2} + 1}{2} = \frac{\sqrt{2} + 1}{4}
\]

Thus, the exact value for part (a) is:
\[
\boxed{\frac{\sqrt{2} + 1}{4}}
\]

---

### Part (b)
\[
\sin\left(\frac{7\pi}{12}\right) + \sin\left(\frac{\pi}{12}\right)
\]

This can be solved using the sum-to-product identity:

\[
\sin(A) + \sin(B) = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
\]

Substituting \(A = \frac{7\pi}{12}\) and \(B = \frac{\pi}{12}\):

\[
\sin\left(\frac{7\pi}{12}\right) + \sin\left(\frac{\pi}{12}\right) = 2 \sin\left(\frac{\frac{7\pi}{12} + \frac{\pi}{12}}{2}\right) \cos\left(\frac{\frac{7\pi}{12} - \frac{\pi}{12}}{2}\right)
\]
\[
= 2 \sin\left(\frac{8\pi}{12} \div 2\right) \cos\left(\frac{6\pi}{12} \div 2\right)
\]
\[
= 2 \sin\left(\frac{4\pi}{12}\right) \cos\left(\frac{3\pi}{12}\right)
\]
\[
= 2 \sin\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right)
\]

Now, use the known values \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\):

\[
= 2 \times \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}
\]
\[
= \frac{\sqrt{6}}{2}
\]

Thus, the exact value for part (b) is:
\[
\boxed{\frac{\sqrt{6}}{2}}
\]

---

Would you like further clarification on these steps?

### Follow-up questions:
1. How are product-to-sum identities derived in trigonometry?
2. What is the geometric significance of the sine and cosine values at standard angles?
3. Can we apply similar identities to expressions involving tangent or cotangent?
4. How do trigonometric identities help in simplifying integrals or derivatives?
5. What is the importance of exact trigonometric values in calculus?

**Tip:** Memorizing exact trigonometric values for common angles (like \(30^\circ\), \(45^\circ\), and \(60^\circ\)) can speed up solving problems significantly.

Find the exact value of each expression: (a) sin(5pi/24) * cos(pi/24), (b) sin(7pi/12) + sin(pi/12)

Learn how to find the exact values of trigonometric expressions using product-to-sum and sum-to-product identities. This problem involves calculating sin(5pi/24) * cos(pi/24) and sin(7pi/12) + sin(pi/12) with step-by-step solutions, illustrating the practical use of these important trigonometric identities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation