Problem 42 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\sin (u+v)+\sin (u-v)=2 \sin u \cos v$$

Step-by-Step Solution

Answer

Verified: \( \sin(u+v) + \sin(u-v) = 2 \sin u \cos v \).

Step 1: Understand the Identity to Verify

We're given the expression \( \sin(u+v) + \sin(u-v) \) and need to verify that it equals \( 2 \sin u \cos v \). To do this, we'll work on simplifying the left-hand side to see if it matches the right-hand side.

Step 2: Use the Sum-to-Product Formulas

Recall the sum-to-product identities: \[ \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right), \]where \( A = u+v \) and \( B = u-v \). Apply this identity to the left-hand side.

Step 3: Simplify Using Specific Values for A and B

Substitute \( A = u+v \) and \( B = u-v \) into the formula:- \( \frac{A+B}{2} = \frac{(u+v) + (u-v)}{2} = u \)- \( \frac{A-B}{2} = \frac{(u+v) - (u-v)}{2} = v \)This simplifies to:\[ 2 \sin(u) \cos(v) \]

Step 4: Confirm that Both Sides Match

After applying the sum-to-product formula, the left-hand side \( \sin(u+v) + \sin(u-v) \) simplifies to \( 2 \sin u \cos v \), which matches the right-hand side exactly.

Key Concepts

Sum-to-Product Formulas

The Sum-to-Product Formulas are an essential tool in trigonometry that help in converting sums of trigonometric functions into products. These formulas come in handy when simplifying expressions or verifying identities. In the context of our exercise, we dealt with the formula for the sum of two sines:

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

This formula is useful because it transforms the addition of two sine terms into a product of sine and cosine, making it easier to evaluate or verify an identity. In our specific example, substituting \( A = u+v \) and \( B = u-v \) allows us to simplify the expression into a format that matches the right-hand side of the given identity.
Understanding how to use these formulas effectively is crucial for mastering problems involving trigonometric identities.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in the study of relationships between angles and sides in right-angled triangles. The sine and cosine functions, denoted by \( \sin \) and \( \cos \), are among the most common and vital functions in trigonometry.
These functions have properties and identities that help in transforming complex expressions or equations. In our example, the functions \( \sin(u+v) \) and \( \sin(u-v) \) represent the sine of the sum and difference of two angles. Both angles \( u \) and \( v \) can be any real numbers or expressions involving other angles.

• Sine Function (\( \sin \)): Represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine Function (\( \cos \)): Represents the ratio of the length of the adjacent side to the hypotenuse.

Understanding these basic functions and their properties is key to solving and simplifying trigonometric identities.

Identity Verification

Identity Verification in trigonometry involves confirming that two expressions are equivalent. It is a critical skill for solving many mathematical problems. The process often uses algebraic manipulation alongside trigonometric formulas like the Sum-to-Product formulas.
In our exercise, we started by rewriting the expression \( \sin(u+v) + \sin(u-v) \). By applying a known identity and simplifying the terms, the expression matched \( 2 \sin u \cos v \), thus verifying the identity.

• Identify the expression to verify.
• Apply relevant trigonometric identities.
• Simplify the expression step-by-step.
• Check if both sides of the equation match.

This step-by-step approach ensures that you correctly verify the identity, providing a solid basis for more complex trigonometric problems. Developing this skill takes practice and a good understanding of various trigonometric identities.