Problem 63 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Write an expression in terms of the integer \(n\) that represents all the \(x\)-intercepts of the graph of the function \(y=\cos 2 x\). Justify your answer.

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=\cos 2x\) are at \( x= (n+0.5)\pi\), where \(n\) is an integer.

Step 1: Understanding Cosine Function

Remember the general form of cosine graph, the x-intercepts of the cosine function are the solutions to \(\cos x = 0\). The cosine function equals zero at \(x= (2n+1) * \frac{\pi}{2}\), for odd integers 'n'.

Step 2: Creating a General Expression

We then substitute this inside the given function (since \( x = n \pi\) in \( \cos 2x\)). Therefore, the function becomes \(\cos(2 * (2n+1)*\frac{\pi}{2} ) = 0\), which simplifies to \( \cos ((2n+1)*\pi) = 0\) .

Step 3: Verifying solution

Now we have the equation \(\cos ((2n+1)\pi) = 0\), which equals zero for all odd integers for \(n\). This can be confirmed by examining the cosine wave graph, which crosses the x-axis at these points. So, the x-intercepts of \(y=\cos 2x\) are given by \( x= (n+0.5)\pi\).

Key Concepts

Trigonometric Functions

Trigonometric functions are the foundations of understanding the relationship between angles and side lengths in right-angled triangles. They are crucial in the study of periodic phenomena such as waves and oscillations. Despite their name suggesting a particular focus on geometry, trigonometric functions indeed bridge realms from geometry to analysis, often encountered in physics and engineering to describe oscillatory and wave-like behaviors.

The primary functions are sine, cosine, and tangent, along with their reciprocals cosecant, secant, and cotangent. The function we're focusing on, the cosine function, is an even function, which means it exhibits symmetry about the y-axis. Its values range from -1 to 1, and it is periodic with a period of \(2\pi\), implying that its values repeat every \(2\pi\) units along the x-axis.

Cosine Graph

The graph of the cosine function is a visualization that shows how the value of cosine changes with respect to an angle measured in radians. It's a smooth, continuous wave that moves above and below the x-axis, touching -1, 0, and 1 at regular intervals. When graphed, the cosine function starts at \(\cos(0)=1\), descends to touch the x-axis at \(\frac{\pi}{2}\), continues to \(\cos(\pi)=-1\), rises back to touch the x-axis at \(\frac{3\pi}{2}\), and completes its cycle at \(\cos(2\pi)=1\).

In the context of the exercise, we are exploring the graph of \(y=\cos 2x\). It's important to note that the '2' in front of the \(x\) compresses the standard cosine graph horizontally by a factor of two, resulting in a function that has a period of \(\pi\) instead of \(2\pi\). Therefore, the x-intercepts occur more frequently—at every half-integer multiple of \(\pi\), to be precise.

X-intercept Expression

The x-intercepts of a graph are the points where the graph crosses the x-axis—where the y-value is zero. For the cosine function, identifying the x-intercepts is based on knowing the angles at which the cosine value is zero. Specifically, for \(\cos 2x = 0\), we need to find the values of \(x\) that make this true.

Since the expression inside the cosine function is \(2x\), we determine where \(\cos (2x)\) equals zero. As we noted from the steps provided, this occurs when \(2x = (2n+1)\frac{\pi}{2}\), where \(n\) is an integer. It's essential to express these x-intercepts in terms of \(n\) to accommodate the infinite nature of the cosine function's x-intercepts. Thus, the x-intercepts are represented by \(x = (n+0.5)\pi\) or, in a more traditional format, as \(x = (2n+1)\frac{\pi}{2}\), indicating that for each odd integer multiple of \(\frac{\pi}{2}\), the cosine graph touches the x-axis.