Problem 31 - 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. $$f(x)=\sin ^{2} x$$

Step-by-Step Solution

Answer

The function \(f(x)=\sin^{2}x\) can be rewritten using the power-reducing identity as \(f(x)=\frac{1-\cos(2x)}{2}\). Then it can be visualized using a graphing utility.

Step 1: Remind of the power-reducing identity

The power-reducing identities in trigonometry for \(\sin^2x\) is \(\frac{1-\cos(2x)}{2}\). This formula helps in reducing the power of a sine function.

Step 2: Applying the identity

Apply the formula on the given function, to reduce its power: \(f(x)=\sin^{2}x=\frac{1-\cos(2x)}{2}\)

Step 3: Graph the resulting function

Now graph the function \(f(x)=\frac{1-\cos(2x)}{2}\) using a graphing utility. The graph will have the same general shape as \(\sin^{2}x\), but it will be compressed horizontally, due to the term \(2x\) in cosine function, and vertically, due to the division by 2. Observe the graph to understand how power reducing formulas can affect the visual representation of a function.

Key Concepts

Trigonometric identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions. They are crucial for simplifying complex trigonometric expressions and solving trigonometric equations. One commonly used identity is the power-reducing formula. It provides a way to express squared trigonometric functions in terms of cosine functions, which are often more straightforward to work with.

For instance, the power-reducing identity for the sine function is:

• \(rac{1 - \cos(2x)}{2}\)

This identity simplifies \(\sin^2x\) by reducing its exponent, transforming it into a form involving just \(\cos(2x)\). Understanding these identities allows mathematicians to handle trigonometric expressions more efficiently, especially when dealing with integration and differentiation in calculus.

Sine function

The sine function, often denoted as \(\sin(x)\), is a fundamental trigonometric function that represents the y-coordinate of a point on the unit circle at an angle \(x\) from the positive x-axis. Its graph is a smooth, wave-like curve, repeating itself every \(2\pi\) units, known as a periodic function.

Key characteristics of the sine function include:

• Amplitude: The height from the centerline of the wave to its peak, typically 1 in the basic sine function.
• Period: The distance over which the wave repeats itself, \(2\pi\) for \(\sin(x)\).
• Symmetry: The sine function is an odd function, meaning \(\sin(-x) = -\sin(x)\).

This function is integral in modeling phenomena such as sound waves and oscillations, where periodic behavior occurs. Understanding the sine function is foundational for mastering more advanced trigonometric concepts and transformations.

Graphing utility

Graphing utilities are tools or software used to visualize mathematical functions. They help in understanding how functions behave visually, which can be essential for analyzing and interpreting mathematical concepts effectively.

When graphing \(f(x) = \frac{1 - \cos(2x)}{2}\) using a graphing utility, it aids in observing the impact of the power-reducing formula on the sine function's graph:

• Horizontal compression: The presence of \(2x\) causes the cosine function, and hence the entire function, to undergo a horizontal compression by a factor of 2.
• Vertical scaling: The division by 2 results in a vertical compression of the function's amplitude.

Using graphing utilities is particularly helpful for students to visualize transformations and understand trigonometric identities, offering insights beyond algebraic manipulation.

Trig functions transformation

Transformations of trigonometric functions involve altering their graphs without changing their fundamental nature. This can include shifts, stretches, and reflections, which modify how the graph appears on the coordinate plane.

For the function \(f(x) = \frac{1 - \cos(2x)}{2}\):

• Horizontal compression occurs due to the \(2x\) term, making the waves of the function appear more frequent within the same interval compared to \(\sin^2x\).
• Vertical scaling results from dividing by 2, reducing the height of the waves, affecting the amplitude.

Understanding these transformations is crucial for graphing and interpreting trigonometric functions correctly. It also plays an important role in real-world applications, such as engineering and physics, where precise modeling of waveforms and oscillations is necessary.