Problem 122 - Algebra and Trigonometry

Question

Use the identity for \(\cos ^{2} x\) to graph one period of \(y=\cos ^{2} x\)

Step-by-Step Solution

Answer

The graph of \(\cos ^{2} x\) is a wave that starts from a maximum at x=0, goes down to 0 at \(x=\frac{\pi}{2}\), returns to 1 at \(x=\pi\), decreases to 0 again at \((3\pi) / 2\), and goes back to 1 at \(x=2\pi\). This cycle repeats for each period of \(2\pi\).

Step 1: Use the Pythagorean identity

Use the Pythagorean identity to rephrase \(\cos ^{2} x\) as \(1- \sin ^{2} x\). This allows an easier plotting of the graph.

Step 2: Find key points

Find the points where the function reaches its maximum and minimum values and the points where the function crosses the x-axis. In the case of \(\cos ^{2} x\), the maximum value is 1, the minimum value is 0, and it crosses the x-axis when \(x = \frac{\pi}{2}\), \((3\pi) / 2\).

Step 3: Plot the function

Use the key points to plot the function. \(\cos ^{2} x\) starts at a maximum at x=0 then drops to 0 at \(x = \frac{\pi}{2}\), goes back up to 1 at \(x=\pi\), drops down to 0 at \((3\pi) / 2\), and goes back to 1 at \(x=2\pi\). Then, connect these plotted points smoothly to create one period of the function

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It connects the squares of sine and cosine functions into a powerful and versatile equation. This identity states:

• \( \sin^2 x + \cos^2 x = 1 \)

This equation holds for any angle \(x\). It is called the "Pythagorean" identity because it is reminiscent of the Pythagorean Theorem, where the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

In this particular exercise, we reformulated \(\cos^2 x\) using this identity. By rearranging, we write \(\cos^2 x = 1 - \sin^2 x\). This form is often more useful, especially when focusing on the cosine squared function, as it integrates smoothly with the graphing process.

Using this expression allows you to more easily determine the behavior of the function, including its symmetry and periodicity, which are particularly important when graphing trigonometric functions.

Graphing Trigonometric Functions

Graphing trigonometric functions is an essential skill in mathematics that helps visualize the behavior of periodic functions. These functions oscillate, producing a repeating wave-like pattern. Here are steps to understand how to graph a trigonometric function:

• Identify Important Points: Determine the points where the function reaches its maximum and minimum values, and where it crosses the x-axis.
• Amplitude and Period: Identify the maximum height (amplitude) and the length of one complete cycle (period) of the graph.
• Symmetry: Recognize any symmetrical properties, which can help make graphing easier.

In the context of the exercise, one full period of \( y = \cos^2 x \) spans from \(x = 0\) to \(x = 2\pi\). The function reaches a maximum value of 1 and a minimum value of 0 within this interval.

Begin plotting these key points and connect them with a smooth curve to reflect the natural oscillation of the cosine squared function. Remember that understanding where the function reaches peak values or zero can greatly enhance your comprehension of its graphical representation.

Cosine Squared Function

The cosine squared function, expressed as \(y = \cos^2 x\), is a variation of the standard cosine function, \(y = \cos x\). This squared function modifies the behavior of the classic cosine wave and makes its properties distinct:

• The range of \(\cos^2 x\) is between 0 and 1, unlike the simple cosine wave which ranges from -1 to 1.
• This function is always non-negative, never dipping below the x-axis since the square of any real number cannot be negative.
• The period of \(\cos^2 x\) is \(\pi\), which is half the period of the standard cosine function (2\(\pi\)).

When plotting \(y = \cos^2 x\), notice how it produces a series of "humps". Each cycle completes within \(\pi\) units along the x-axis, contrasting with the typical wave-like cosine function that spans 2\(\pi\). This squaring effect causes each period to involve segments where the curve touches the x-axis (at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\)), creating a distinct pattern helpful in visualizing varied applications in trigonometry, physics, and engineering.