Problem 66 - Algebra and Trigonometry

Question

\(61-66\) Write the sum as a product. $$ \sin 3 x+\sin 4 x $$

Step-by-Step Solution

Answer

\( 2 \sin \left( \frac{7x}{2} \right) \cos \left( \frac{x}{2} \right) \)

Step 1: Identify the Trigonometric Identity

The expression can be rewritten using the sum-to-product identities for sine. The identity we need is: \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \). Here, \( A = 3x \) and \( B = 4x \).

Step 2: Calculate \( A + B \)

Using the given values, calculate \( A + B = 3x + 4x = 7x \).

Step 3: Calculate \( \frac{A + B}{2} \)

Divide the sum by 2: \( \frac{7x}{2} \).

Step 4: Calculate \( A - B \)

Using the given values, calculate \( A - B = 3x - 4x = -x \).

Step 5: Calculate \( \frac{A - B}{2} \)

Divide the difference by 2: \( \frac{-x}{2} = -\frac{x}{2} \).

Step 6: Apply the Identity

Substitute \( \frac{A + B}{2} \) and \( \frac{A - B}{2} \) into the identity: \( \sin 3x + \sin 4x = 2 \sin \left( \frac{7x}{2} \right) \cos \left( -\frac{x}{2} \right) \).

Step 7: Simplify the Expression

Since \( \cos(-\theta) = \cos(\theta) \), the expression simplifies to: \( 2 \sin \left( \frac{7x}{2} \right) \cos \left( \frac{x}{2} \right) \).

Key Concepts

Trigonometric Identities

In trigonometry, identities are equations that hold for all values of the variables involved, usually incorporating trigonometric functions like sine, cosine, and tangent. One crucial category of these identities is the sum-to-product identities, which allow us to rewrite the sum or difference of two trigonometric functions as a product. This can simplify calculations and make solving trigonometric equations more efficient.

• **Sum-to-product identity for sine:** If we take the example of sine, the identity for the sum of sine functions is given by:\[\sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\]
• These identities are particularly useful in algebraic transformations and simplifying expressions for further calculations.

By breaking down the angles and expressing the sum of sine functions as a product of sine and cosine, we can delve deeper into trigonometric problems with ease.

Sine Function

The sine function, commonly denoted as \(\sin\), is one of the primary trigonometric functions. It is widely used in various mathematical applications, especially in the analysis of periodic phenomena such as sound and light waves. The sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

• **Periodicity of sine:** The sine function has a periodic nature with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
• **Properties:** It's important to note that the sine function is odd, satisfying the identity \(\sin(-x) = -\sin(x)\).
• **Amplitude and frequency:** The function can be modified to change its amplitude and frequency, leading to variations such as \(\sin(kx)\) where \(k\) affects these parameters.

Understanding these properties allows us to manipulate and work with the sine function effectively in mathematical exercises and real-world applications.

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions in different forms to simplify calculations and solving problems. This skill is fundamental in mathematics and helps in converting complex equations into more manageable forms.

Simplification Process

The process often includes finding common factors, expanding expressions, or applying known identities. For trigonometric expressions, this might mean:

• Using trigonometric identities to rewrite terms
• Simplifying by canceling common factors
• Transforming double angles, half angles, or converting sums into products

Applying these techniques carefully can streamline the solving process. In our exercise, transforming \(\sin 3x + \sin 4x\) involves:

• Calculating \(A + B\) and \(\frac{A + B}{2}\)
• Calculating \(A - B\) and \(\frac{A - B}{2}\)
• Substituting these into the identity to simplify the expression

By understanding the principles of algebraic manipulation, students can convert intricate trigonometric expressions into simpler forms to solve them easily.