Problem 2 - Algebra 2 and Trigonometry

Question

Aaron solved the equation \(2 \sin \theta \cos \theta=\cos \theta\) by first dividing both sides of the equation by cos \(\theta\) . Aaron said that for \(0 \leq \theta \leq 2 \pi,\) the solution set is \(\left\\{\frac{\pi}{6}, \frac{5 \pi}{6}\right\\} .\) Do you agree with Aaron? Explain why or why not.

Step-by-Step Solution

Answer

Aaron's solution is incomplete; missing solutions are \(\theta = \frac{\pi}{2}\) and \(\frac{3\pi}{2}\).

Step 1: Analyze the Given Equation

The given equation is \(2 \sin \theta \cos \theta = \cos \theta\). Aaron divided both sides by \(\cos \theta\), but this implies that \(\cos \theta eq 0\) to avoid division by zero.

Step 2: Solve for Cos Theta Equaling Zero

Identify when \(\cos \theta = 0\), because dividing by zero is undefined. This occurs at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\) within the interval \([0, 2\pi]\). These must be checked separately.

Step 3: Divide and Simplify the Equation

Assuming \(\cos \theta eq 0\), divide the entire equation by \(\cos \theta\): \[2 \sin \theta \cos \theta / \cos \theta = \cos \theta / \cos \theta\] simplifies to \[2 \sin \theta = 1\] which gives \(\sin \theta = \frac{1}{2}\).

Step 4: Solve \(\sin \theta = \frac{1}{2}\)

Determine the angles for which \(\sin \theta = \frac{1}{2}\) within the interval \([0, 2\pi]\). The values of \(\theta\) that satisfy this are \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\).

Step 5: Check Solutions at \(\theta = \frac{\pi}{2}\) and \(\frac{3\pi}{2}\)

Since these points make \(\cos \theta = 0\), substitute them back into the original equation:\(2 \sin \frac{\pi}{2} \cdot 0 = 0\) and \(2 \sin \frac{3\pi}{2} \cdot 0 = 0\) are both true. Thus, these points are also solutions.

Key Concepts

Division by Zero

In mathematics, division by zero is a concept that is undefined. This means you cannot divide any number by zero because it does not produce a meaningful or finite result. When solving the trigonometric equation \(2 \sin \theta \cos \theta = \cos \theta\), one might be tempted to divide both sides by \(\cos \theta\). However, this action is only valid if \(\cos \theta eq 0\).

If \(\cos \theta = 0\), then dividing by \(\cos \theta\) would lead to division by zero, which is not allowed. This step in solving the equation needs careful consideration to ensure no mistakes are made. It's important to check and handle situations where \(\cos \theta = 0\) separately by directly evaluating the original equation for these specific cases.

Interval [0, 2π]

The interval \([0, 2\pi]\) represents all the possible values of \(\theta\) within one complete cycle of a circle in radians. When working with trigonometric functions, it's essential to consider this interval to find all solutions that are relevant to the situation.

For example, in the given problem, we need to find all the solutions of the equation \(2 \sin \theta \cos \theta = \cos \theta\) within this interval. This range includes all angles from 0 to \(2\pi\) (or 0 to 360 degrees), which encompasses all standard positions where trigonometric values are repeatedly evaluated, ensuring no possible solutions are left out.

Sine and Cosine Values

Sine and cosine are fundamental trigonometric functions that describe a relationship between angles and ratios of sides in right triangles. These functions are also central in describing circular motion.

For sine, \(\sin \theta = \frac{1}{2}\) is a well-known value. Within the interval \([0, 2\pi]\), \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\) are the specific angles that satisfy this relationship. The cosine function is zero at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), which also require consideration as these could lead to potential solutions where division by zero might have been originally overlooked.

Understanding these values and their corresponding angles within different intervals helps solve trigonometric equations accurately.

Solution Verification

Solution verification is an essential step in solving equations, particularly when initial assumptions could exclude certain possibilities. Once solutions to the equation \(2 \sin \theta \cos \theta = \cos \theta\) are found, each one should be checked within the context of the original problem.

Rechecking the solutions involves plugging the found values back into the original equation to ensure they hold true. For this problem, substituting \(\theta = \frac{\pi}{6}\), \(\theta = \frac{5\pi}{6}\), \(\theta = \frac{\pi}{2}\), and \(\theta = \frac{3\pi}{2}\) back into the equation ensures all potential solutions, including those that might lead to division by zero, are accounted for. Note that both \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\) make the term \(\cos \theta = 0\), confirming their validity if they result in a true statement when verified against the original equation.