Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\sin 2 x+\cos x=0$$

Step-by-Step Solution

Answer

The exact solutions of the equation \(\sin 2x + \cos x = 0\) in the interval \([0,2\pi)\) are \(x = \pi/2, 3\pi/2, 5\pi/6, 11\pi/6\).

Step 1: Apply the Double-Angle Identity

Given the equation \(\sin 2x + \cos x = 0\). We know the double angle identity of sine which is \(\sin 2x = 2 \sin x \cos x\). We substitute this in place of \(\sin 2x\) in the equation to get \(2 \sin x \cos x + \cos x = 0\).

Step 2: Factor out the Common Factor

In equation \(2 \sin x \cos x + \cos x = 0\), it can be observed that \(cos x\) is a common factor, so it can be factored out to simplify the equation. The factored equation becomes \(\cos x (2 \sin x + 1) = 0\).

Step 3: Set Each Factor to Zero

The next step is to solve the equation \(\cos x (2 \sin x + 1) = 0\) by setting each factor to zero, which leads to \(\cos x = 0\) or \(2 \sin x + 1 = 0\).

Step 4: Solve the Equations

Solving the equations \(\cos x = 0\) and \(2 \sin x + 1 = 0\) gives values for \(x\). For the equation \(\cos x = 0\), \(x\) is \(\pi/2\) or \(3\pi/2\). And for the equation \(2 \sin x + 1 = 0\), solving for \(x\) gives \(x = \sin^{-1}(-1/2)\), which means \(x\) is \(-\pi/6\) or \(-5\pi/6\). However, since we are looking for solutions in the interval \([0,2 \pi)\) , these negative solutions would become \(5\pi/6\) and \(11\pi/6\) respectively.

Step 5: Graphical Solution

To further confirm these solutions, you can plot the graphs of \(\sin 2x\) and \(-\cos x\). Where these graphs intersect on the interval \([0,2 \pi)\) would be the approximate solutions to the original equation.

Key Concepts

Double-Angle Identity

Understanding the double-angle identities is crucial in solving trigonometric equations efficiently. The double-angle identity for sine states that \(\sin(2x) = 2\sin(x)\cos(x)\). This relationship is derived from the sum of angles formula and essentially says that the sine of an angle that is double another angle can be expressed using the sine and cosine of just the original angle. It's like having a mathematical shortcut.

When we apply the double-angle identity, as seen in the given problem, it helps to transform the equation into a more solvable form. By replacing \(\sin(2x)\) with \(2\sin(x)\cos(x)\), we're setting the stage to factor the equation, which is a common next step in finding solutions to trigonometric equations.

Factoring Trigonometric Expressions

Factoring is a powerful tool for solving equations in general, and it applies to trigonometric expressions as well. When we're presented with a trigonometric expression like \(2\sin(x)\cos(x) + \cos(x) = 0\), identifying common factors is key. In this case, \(\cos(x)\) is a common factor.

Here's a breakdown of the factoring process:

• First, look at each term in the equation and find any common trigonometric functions.
• Next, factor out the common trigonometric function from all the terms.
• What you’re left with is a simpler way to look at the equation, split into two parts that can be solved individually.

The act of factoring simplifies the problem and reveals simpler paths to reach the solution, allowing for the trigonometric equation to be split into more manageable pieces that can be solved algebraically.

Graphing Utility in Trigonometry

Graphing utilities are not just a way to visually represent equations, they are extremely useful in confirming the solutions we find algebraically. They especially come in handy when dealing with trigonometric equations where the intersection of graphs can indicate solutions.

By graphing the functions \(\sin(2x)\) and \(\cos(x)\), and looking for where they intersect within a specific interval such as \(0, 2\pi\), we can pinpoint the approximate solutions. Graphing adds another layer to our understanding, for sometimes seeing is believing—and corroborating our solutions graphically strengthens our confidence in them.

Moreover, with the graph, we can visualize the periodic nature of trigonometric functions, see patterns, and understand the solutions in the context of these patterns. Graphing utilities not only assist in solving the problem but also enrich our comprehension of trigonometric behavior over different intervals.