Problem 33 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\sin (x+y)-\sin (x-y)=2 \cos x \sin y$$

Step-by-Step Solution

Answer

The identity is proven using sine addition and subtraction formulas.

Step 1: Recall the Sine Addition and Subtraction Formulas

The sine addition formula is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) and the sine subtraction formula is \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). We'll use these to expand the given identity.

Step 2: Expand \( \sin(x + y) \)

Using the sine addition formula, \( \sin(x + y) = \sin x \cos y + \cos x \sin y \).

Step 3: Expand \( \sin(x - y) \)

Using the sine subtraction formula, \( \sin(x - y) = \sin x \cos y - \cos x \sin y \).

Step 4: Subtract \( \sin(x - y) \) from \( \sin(x + y) \)

Subtracting the two expansions, we have: \[\sin(x+y) - \sin(x-y) = (\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y)\]

Step 5: Simplify the Expression

Combine like terms:\[ (\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y) = 2 \cos x \sin y \]

Step 6: Conclusion

We have shown that substituting the known formulas and simplifying the expression results in the desired identity.Thus, \( \sin(x+y) - \sin(x-y) = 2\cos x \sin y \) is proven to be true.

Key Concepts

Sine Addition Formula

The Sine Addition Formula is a cornerstone of trigonometry. It helps us find the sine of the sum of two angles. When you want to calculate the sine of the angles added together, such as \( a + b \), the formula is: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). This property gives us a convenient way to express and manipulate trigonometric expressions with angle sums.

This formula is incredibly useful because it allows us to convert more complex expressions into simpler ones using basic trigonometric functions. If you are working with angles where individually their sine and cosine values are known, using this formula can turn a daunting expression into a manageable calculation.

• \( \sin \) and \( \cos \) are trigonometric functions representing sine and cosine, respectively.
• \( a \) and \( b \) are the angles being added together.
• Helps to transform and simplify trigonometric expressions.

Sine Subtraction Formula

Just like the addition formula, the Sine Subtraction Formula allows us to find the sine of the difference between two angles. The expression is \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). This formula is similar to the addition formula, but with a critical subtraction in between terms.

Utilizing the Sine Subtraction Formula, you can transform problems involving angle differences into a combination of simpler trigonometric functions. This method is highly practical when solving trigonometric identities or equations.

• \( \sin \) signifies sine, while \( \cos \) is for cosine.
• \( a \) and \( b \) represent two different angles.
• Key for breaking down and solving complex problems involving angle differences.

Trigonometric Proofs

Trigonometric proofs are fascinating and crucial in verifying the equality of trigonometric expressions. They involve showing that one side of an equation can be transformed into the other through logical steps, often using known identities, like the Sine Addition and Subtraction Formulas.

In our provided problem, proving the identity \( \sin(x+y) - \sin(x-y) = 2\cos x \sin y \) is a classic example. The approach involves: breaking down the equations into simpler components using sine formulas, simplifying the expressions, and showing they match the given identity.

The main objective in trigonometric proofs is to ensure clarity and precision in each step, ensuring the logic is sound and each transformation follows directly from established identities.

• Utilize known identities like the Sine Addition and Subtraction Formulas.
• Simplify expressions to show they match the problem statement.
• Ensure each step logically follows the previous one.