Problem 69 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \sin(x + y) + \sin(x - y) = 2 \sin x \cos y \)

Step-by-Step Solution

Answer

The identity \( \sin(x + y) + \sin(x - y) = 2 \sin x \cos y \) is successfully proven.

Step 1: Express sin(a + b) and sin(a - b) in terms of sin and cos

The formulas for \(\sin(a + b)\) and \(\sin(a - b)\) are known. They are \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) and \(\sin(a - b) = \sin a \cos b - \cos a \sin b\). Here, replace \(a\) with \(x\) and \(b\) with \(y\) in these formulas.

Step 2: Substitute the formulas into the equation

Now, substitute \(\sin(x + y)\) and \(\sin(x - y)\) in the given identity with the formulas derived in Step 1. This will give: \((\sin x \cos y + \cos x \sin y)+(\sin x \cos y - \cos x \sin y)\).

Step 3: Simplify the equation

By combining like terms, the equation simplifies to: \(2 \sin x \cos y\).

Step 4: Conclude the proof

The resulting expression is the same as the right-hand side of the original identity, thus proving the identity is true.

Key Concepts

Sine Addition Formula

When looking at the sine addition formula, it's about understanding how the sine of a sum of two angles relates to their individual sines and cosines. Concretely, the formula is expressed as:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \].

This formula is crucial for solving various trigonometric problems, including our exercise where we have to prove an identity involving \( \sin(x + y) + \sin(x - y) \). The sine addition formula helps break down the sine of a sum into more manageable parts. This is particularly useful when angles are given in terms of variables, as is often the case in algebra.

Cosine Subtraction Formula

The cosine subtraction formula mirrors the sine addition formula and is another tool for deconstructing the trigonometric functions of combined angles. In contrast to sine, the cosine subtraction looks like:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \].

Although not directly used in our exercise, the cosine subtraction formula is a counterpart to the sine formulas mentioned in the solution steps. Understanding this identity assists in comprehending the broader range of trigonometric identities that often arise in more complex problems.

Proving Trigonometric Identities

Proving trigonometric identities is a cornerstone exercise in mastering trigonometry. It involves confirming that two different trigonometric expressions are equivalent for all values of the included variables. The process, as shown in our exercise, consists of a few steps:

1. Begin with known trigonometric identities, like sine and cosine addition formulas.
2. Replace the generalized angles in these formulas with the specific variables given in your exercise.
3. Substitute these expressions into your equation and simplify where possible, combining like terms to reveal the identity.
4. Finally, compare the simplified expression with the other side of your equation. If they match, the proof is successful.

The ability to prove identities strengthens comprehension of key trigonometric concepts and is a valuable skill in fields that employ mathematical reasoning.

Trigonometric Functions

Let's delve into the world of trigonometric functions. These are fundamental to trigonometry and are functions of an angle. The main functions include sine (sin), cosine (cos), and tangent (tan), each of which represents a ratio of sides within a right-angled triangle. Here's a basic overview:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

Understanding these ratios provides the backbone for working with problems involving angles, whether you're dealing with right triangles or unit circles. These functions are also periodic, meaning they repeat values over intervals, and understanding this concept is essential for dealing with trigonometric equations and identities.