Problem 35 - Algebra and Trigonometry

Question

Mixed Practice In Problems \(35-40,\) (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ f(x)=\sin (2 x) \sin x $$

Step-by-Step Solution

Answer

\(\sin (2x) \sin (x) = \frac{1}{2} \cos (x) - \frac{1}{2} \cos (3x)\)

Step 1: Identify the appropriate Product-to-Sum Formula

The Product-to-Sum Formulas for sine are: oindent \(\sin A \sin B = \frac{1}{2} [\cos (A-B) - \cos (A+B)]\). For this problem, identify A and B: \(A = 2x\) and \(B = x\).

Step 2: Apply the Product-to-Sum Formula

Using the values of \(A = 2x\) and \(B = x\) in the formula \(\sin A \sin B = \frac{1}{2} [\cos (A-B) - \cos (A+B)]\), we get: oindent\(\sin (2x) \sin (x) = \frac{1}{2} [\cos (2x - x) - \cos (2x + x)]\). Simplify the expression inside the cosine functions: oindent \(\sin (2x) \sin (x) = \frac{1}{2} [\cos (x) - \cos (3x)]\).

Step 3: Write the function as a sum

Based on the previous step, \(\sin (2x) \sin (x)\) can be rewritten as: oindent \(\sin (2x) \sin (x) = \frac{1}{2} \cos (x) - \frac{1}{2} \cos (3x)\). This is the required expression of the product as a sum.

Step 4: Graph the function

To graph the function \(f(x) = \frac{1}{2} \cos (x) - \frac{1}{2} \cos (3x)\), first evaluate the function at key points (e.g., \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)). Plot these points and draw the curve based on these points for the interval \( [0, 2\pi]\). Remember, you can use the method of adding \(y\)-coordinates for more accurate plotting.

Key Concepts

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are used to simplify complex trigonometric expressions and solve equations. Examples include the Pythagorean identity, the angle sum and difference identities, and the product-to-sum formulas. These identities are essential tools in trigonometry and calculus. For instance, the product-to-sum formulas help transform products of trigonometric functions into sums, making integration and differentiation easier.

graphing trigonometric functions

Graphing trigonometric functions involves plotting sine, cosine, tangent, and other trigonometric functions over a specified interval. The key steps include:

• Identifying the function's amplitude, period, and phase shift
• Calculating key points, such as maximum, minimum, and intercepts
• Plotting these points and connecting them smoothly

For the function \( f(x) = \frac{1}{2} \cos(x) - \frac{1}{2} \cos(3x) \), plot points at key angles like 0, π/2, π, 3π/2, and 2π. Graphing helps visualize the behavior of trigonometric functions over intervals and understand their properties.

product-to-sum formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences. This makes certain integrals and algebraic manipulations easier. The formula for the product of sins is:\[ \sin(A) \sin(B) = \frac{1}{2} [ \cos(A-B) - \cos(A+B) ] \] By identifying A and B correctly, we can apply this transformation, as in: \( A = 2x \) and \( B = x \) \[ \sin(2x) \sin(x) = \frac{1}{2}[ \cos(x) - \cos(3x) ] \]

sine and cosine functions

Sine and cosine functions are fundamental trigonometric functions. The sine function,\( \sin(x) \), and the cosine function, \( \cos(x) \), both have ranges from -1 to 1 and periods of \( 2π \). They are used to model oscillatory behaviors, such as sound waves, light waves, and alternating currents. Understanding their properties, such as amplitude (the height of peaks), period (time to complete one cycle), and phase shift (horizontal shift), is crucial for solving trigonometric problems.

interval [0, 2π]

The interval [0, 2π] represents one complete cycle of sine and cosine functions. Within this interval:

• \( \sin(x) \) starts at 0, reaches 1 at \( \frac{π}{2}\), returns to 0 at \( π \), drops to -1 at \( \frac{3π}{2}\), and cycles back to 0 at \( 2π \).
• \( \cos(x) \) starts at 1, drops to 0 at \( \frac{π}{2}\), dips to -1 at \( π \), rises back to 0 at \( \frac{3π}{2}\), and returns to 1 at \( 2π \).

This interval is key for understanding the periodic nature and overall behavior of trigonometric functions.