Evaluate The Expression: \sin \frac{\pi}{4} \sin \frac{\pi}{6} = \frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right ]Provide Your Answer Here:

Mar 12, 2025 by ADMIN 155 views

**Evaluating Trigonometric Expressions: A Step-by-Step Guide**

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Introduction

Trigonometric expressions are an essential part of mathematics, and evaluating them requires a deep understanding of the underlying concepts. In this article, we will focus on evaluating the expression $\sin \frac{\pi}{4} \sin \frac{\pi}{6} = \frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)$ . We will break down the solution into manageable steps, using a combination of trigonometric identities and formulas to simplify the expression.

Understanding the Problem

The given expression involves the product of two sine functions, $\sin \frac{\pi}{4}$ and $\sin \frac{\pi}{6}$ . Our goal is to simplify this expression and rewrite it in the form $\frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)$ . To do this, we will need to use various trigonometric identities and formulas.

Step 1: Apply the Product-to-Sum Formula

The product-to-sum formula states that $\sin A \sin B = \frac{1}{2} \left[\cos(A-B) - \cos(A+B)\right]$ . We can apply this formula to the given expression by setting $A = \frac{\pi}{4}$ and $B = \frac{\pi}{6}$ .

import math
angle_A = math.pi / 4
angle_B = math.pi / 6

result = 0.5 * (math.cos(angle_A - angle_B) - math.cos(angle_A + angle_B))

Step 2: Simplify the Expression

Using the product-to-sum formula, we can rewrite the expression as $\frac{1}{2} \left[\cos\left(\frac{\pi}{4} - \frac{\pi}{6}\right) - \cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right)\right]$ . Simplifying the angles inside the cosine functions, we get $\frac{1}{2} \left[\cos\left(\frac{\pi}{12}\right) - \cos\left(\frac{5\pi}{12}\right)\right]$ .

Step 3: Apply the Sum-to-Product Formula

The sum-to-product formula states that $\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$ . We can apply this formula to the expression by setting $A = \frac{\pi}{12}$ and $B = \frac{5\pi}{12}$ .

# Apply the sum-to-product formula
result = -1 * (2 * math.sin((angle_A + angle_B) / 2) * math.sin((angle_A - angle_B) / 2))

Step 4: Simplify the Expression

Using the sum-to-product formula, we can rewrite the expression as $-2 \sin\left(\frac{\frac{\pi}{12} + \frac{5\pi}{12}}{2}\right) \sin\left(\frac{\frac{\pi}{12} - \frac{5\pi}{12}}{2}\right)$ . Simplifying the angles inside the sine functions, we get $-2 \sin\left(\frac{\pi}{3}\right) \sin\left(-\frac{\pi}{12}\right)$ .

Step 5: Evaluate the Sine Functions

Using the values of $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ and $\sin\left(-\frac{\pi}{12}\right) = -\sin\left(\frac{\pi}{12}\right)$ , we can rewrite the expression as $-2 \cdot \frac{\sqrt{3}}{2} \cdot -\sin\left(\frac{\pi}{12}\right)$ .

Step 6: Simplify the Expression

Simplifying the expression, we get $\sqrt{3} \sin\left(\frac{\pi}{12}\right)$ .

Step 7: Evaluate the Sine Function

Using the value of $\sin\left(\frac{\pi}{12}\right) = \sin\left(\frac{\pi}{3} - \frac{\pi}{4}\right)$ , we can rewrite the expression as $\sqrt{3} \sin\left(\frac{\pi}{3} - \frac{\pi}{4}\right)$ .

Step 8: Apply the Angle Subtraction Formula

The angle subtraction formula states that $\sin(A-B) = \sin A \cos B - \cos A \sin B$ . We can apply this formula to the expression by setting $A = \frac{\pi}{3}$ and $B = \frac{\pi}{4}$ .

# Apply the angle subtraction formula
result = math.sqrt(3) * (math.sin(angle_A) * math.cos(angle_B) - math.cos(angle_A) * math.sin(angle_B))

Step 9: Simplify the Expression

Using the angle subtraction formula, we can rewrite the expression as $\sqrt{3} \left(\sin\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{4}\right)\right)$ .

Step 10: Evaluate the Sine and Cosine Functions

Using the values of $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ , $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ , $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ , and $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ , we can rewrite the expression as $\sqrt{3} \left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}\right)$ .

Step 11: Simplify the Expression

Simplifying the expression, we get $\sqrt{3} \left(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\right)$ .

Step 12: Combine the Terms

Combining the terms, we get $\sqrt{3} \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)$ .

Step 13: Rationalize the Denominator

Rationalizing the denominator, we get $\frac{\sqrt{18} - \sqrt{6}}{4}$ .

Step 14: Simplify the Expression

Simplifying the expression, we get $\frac{3\sqrt{2} - \sqrt{6}}{4}$ .

Step 15: Rewrite the Expression

Rewriting the expression, we get $\frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)$ .

The final answer is: $\boxed{\frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)}$

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Introduction

In our previous article, we evaluated the expression $\sin \frac{\pi}{4} \sin \frac{\pi}{6} = \frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)$ . In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in evaluating trigonometric expressions.

Q: What is the product-to-sum formula?

A: The product-to-sum formula is a trigonometric identity that states $\sin A \sin B = \frac{1}{2} \left[\cos(A-B) - \cos(A+B)\right]$ . This formula can be used to simplify the product of two sine functions.

Q: How do I apply the product-to-sum formula?

A: To apply the product-to-sum formula, you need to set $A = \frac{\pi}{4}$ and $B = \frac{\pi}{6}$ in the given expression. Then, you can use the formula to rewrite the expression as $\frac{1}{2} \left[\cos\left(\frac{\pi}{4} - \frac{\pi}{6}\right) - \cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right)\right]$ .

Q: What is the sum-to-product formula?

A: The sum-to-product formula is a trigonometric identity that states $\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$ . This formula can be used to simplify the difference of two cosine functions.

Q: How do I apply the sum-to-product formula?

A: To apply the sum-to-product formula, you need to set $A = \frac{\pi}{12}$ and $B = \frac{5\pi}{12}$ in the given expression. Then, you can use the formula to rewrite the expression as $-2 \sin\left(\frac{\frac{\pi}{12} + \frac{5\pi}{12}}{2}\right) \sin\left(\frac{\frac{\pi}{12} - \frac{5\pi}{12}}{2}\right)$ .

Q: What is the angle subtraction formula?

A: The angle subtraction formula is a trigonometric identity that states $\sin(A-B) = \sin A \cos B - \cos A \sin B$ . This formula can be used to simplify the difference of two sine functions.

Q: How do I apply the angle subtraction formula?

A: To apply the angle subtraction formula, you need to set $A = \frac{\pi}{3}$ and $B = \frac{\pi}{4}$ in the given expression. Then, you can use the formula to rewrite the expression as $\sqrt{3} \left(\sin\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{4}\right)\right)$ .

Q: What is the final answer?

A: The final answer is $\frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)$ .

Conclusion

Evaluating trigonometric expressions requires a deep understanding of the underlying concepts and formulas. By applying the product-to-sum formula, sum-to-product formula, angle subtraction formula, and other trigonometric identities, you can simplify complex expressions and arrive at the final answer. We hope this Q&A guide has helped you understand the concepts and steps involved in evaluating trigonometric expressions.

Frequently Asked Questions

Q: What is the product-to-sum formula?

A: The product-to-sum formula is a trigonometric identity that states $\sin A \sin B = \frac{1}{2} \left[\cos(A-B) - \cos(A+B)\right]$ .

Q: How do I apply the product-to-sum formula?

A: To apply the product-to-sum formula, you need to set $A = \frac{\pi}{4}$ and $B = \frac{\pi}{6}$ in the given expression.

Q: What is the sum-to-product formula?

A: The sum-to-product formula is a trigonometric identity that states $\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$ .

Q: How do I apply the sum-to-product formula?

A: To apply the sum-to-product formula, you need to set $A = \frac{\pi}{12}$ and $B = \frac{5\pi}{12}$ in the given expression.

Q: What is the angle subtraction formula?

A: The angle subtraction formula is a trigonometric identity that states $\sin(A-B) = \sin A \cos B - \cos A \sin B$ .

Q: How do I apply the angle subtraction formula?

A: To apply the angle subtraction formula, you need to set $A = \frac{\pi}{3}$ and $B = \frac{\pi}{4}$ in the given expression.

Q: What is the final answer?

A: The final answer is $\frac{1}{2} \left(-\frac{\pi}{12} - \cos \frac{5\pi}{12}\right)$ .