Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the function using the power-reducing formulas. Then use a graphing utility to graph the function. $$f(x)=\sin ^{3} x$$

Step-by-Step Solution

Answer

The function \(f(x)=\sin ^{3} x\) can be rewritten using the power-reducing formula as: \(f(x) = \frac{\sin x - \sin x \cdot \cos(2x)}{2}\). The graph of this function can be plotted using any graphing utility.

Step 1: Apply the Power-Reducing Formula

Power-reducing formula is particularly useful in this task. The formula is \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\). Notice that \(\sin^3 x = \sin x \cdot \sin^2 x\). So, substitute power reducing identity into sin^2(x) in the function \(\sin^3 x\).\nWe obtain:\[f(x) = \sin x \cdot \frac{1 - \cos(2x)}{2}\]

Step 2: Simplify the Equation

Now, simplify the equation as follows:\[f(x) = \frac{\sin x - \sin x \cdot \cos(2x)}{2}\]

Step 3: Graph the Function

Plot this equation using a graphing utility. On most graphing calculators, this will involve inputting the equation into the y= section, then adjusting the viewing window to get the best view of the curve. Remember that the x-axis represents the 'x' and the y-axis represents 'f(x)', and plot accordingly.

Key Concepts

Understanding the Sine Function

Let's explore the sine function, which is one of the primary trigonometric functions. Represented as \( \text{sin}(x) \), the sine function maps the angle \( x \) (often measured in radians) to the y-coordinate of the point on the unit circle corresponding to that angle.

The sine function has some distinctive characteristics: it is periodic with a period of \( 2\text{π} \), has a range from -1 to 1, and is oscillatory, meaning it repeats values in a wave-like manner. Knowing these properties is crucial when graphing the sine function since it dictates its basic shape and direction of oscillation.

Graphically, if you plot the sine function with an angle on the x-axis and the sine value on the y-axis, you'll get an undulating curve that arises from the unit circle's y-coordinate. This curve is the graphical representation of the sine wave, frequently used in both mathematical analysis and practical applications like sound waves or alternating current (AC) in electricity.

Graphing Trigonometric Functions

Graphing trigonometric functions, like the sine function, requires an understanding of their periodicity, amplitude, frequency, and phase shift. To begin with, the periodicity is the interval after which the function begins to repeat its values – for sine and cosine, this is \( 2\text{π} \).

The amplitude determines the function's peak value, while frequency relates to the number of cycles the function completes in a given interval. Phase shift indicates the horizontal shift of the graph from its standard position.

When graphing \( f(x) = \text{sin}^3(x) \), you'll first need to depict the basic sin(x) wave, and then adjust it according to the transformations caused by the exponentiation to the third power. Transformations could include vertical stretching and more complex changes to the wave's shape. A graphing utility software can handle these transformations, making it easier to visualize the modified curve. To get the best view of the function's behavior, adjust the viewing window to an appropriate scale, and note critical features like intercepts, maximum and minimum points, and intervals of increase/decrease.

Trigonometric Identities and Power-Reducing Formulas

Trigonometric identities are equations involving trigonometric functions that are true for all values within their domains. These identities are the tools that offer shortcuts in solving trigonometry problems and simplifying expressions. One subset of these identities is the power-reducing formulas, which allow the simplification of higher powers of trigonometric functions into first-degree expressions.

For instance, the power-reducing formula for the sine function is given by \( \text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2} \). This can be incredibly useful when you encounter a higher power of sine, such as \( \text{sin}^3(x) = \text{sin}(x) \times \text{sin}^2(x) \), allowing you to reduce the power and simplify the expression into something more manageable for both analysis and graphing.

Using power-reducing formulas not only assists in simplification but also aids in revealing fundamental properties of the function, such as periodicity and symmetry, which might not have been obvious in their original form.