Problem 140 - Precalculus Essentials

Question

In the interval \([0,2 \pi),\) the solutions of \(\sin x=\cos 2 x\) are \(\frac{\pi}{6}, \frac{5 \pi}{6},\) and \(\frac{3 \pi}{2} .\) Explain how to use graphs generated by a graphing utility to check these solutions.

Step-by-Step Solution

Answer

By graphing both \(\sin x\) and \(\cos 2x\) on the same plot, the intersection points, which represent the solutions to the equation \(\sin x=\cos 2x\), can be visually confirmed. From the graphing process, it is evident that the intersection points are indeed at \(\frac{\pi}{6}, \frac{5 \pi}{6}\), and \(\frac{3 \pi}{2}\), corroborating the given solutions.

Step 1: Graph the first function

Begin by plotting the function \(\sin x\) over the interval [0,2π). Since sine is a common trigonometric function, it should yield a wave-like pattern with peaks at \(\frac{\pi}{2}\) and \(\frac{3 \pi}{2}\), crossing the x-axis at \(0, \pi, 2 \pi\).

Step 2: Graph the second function

Next, plot the function \(\cos 2x\) over the same interval [0,2π). The \(\cos 2x\) function will complete two full cycles within this interval, meaning the graph will go through two complete wave patterns from peak to trough.

Step 3: Identify the intersecting points

With both functions graphed, observe where the two curves intersect i.e. where \(\sin x\) equals to \(\cos 2x\). Plotting such coordinates would represent the solution set of the equation. If the graphs intersect at \(\frac{\pi}{6}, \frac{5 \pi}{6}\), and \(\frac{3 \pi}{2}\), then this visually confirms the given solutions. Be mindful that the intersection points should match the x coordinates of the solutions. The y coordinates are simply the values of the functions at those points and should be equal due to the premise of the equation (i.e. \(\sin x=\cos 2x\)).