Problem 1 - Algebra 2 and Trigonometry

Question

Explain why the solution set of the equation \(2 x+4=8\) is \(\\{2\\}\) but the solution set of the equation \(2 \sin x+4=8\) is \(\\{ \\},\) the empty set.

Step-by-Step Solution

Answer

The solution set for \(2x+4=8\) is \(\{2\}\), whereas for \(2\sin x+4=8\), it's \(\{ \}\) since \(\sin x\) cannot equal 2.

Step 1: Simplify Equation 1

Start by simplifying the first equation: \[2x + 4 = 8\]Subtract 4 from both sides to isolate terms with the variable:\[2x + 4 - 4 = 8 - 4\]This simplifies to:\[2x = 4\]

Step 2: Solve for x in Equation 1

Next, divide both sides of the simplified equation by 2 to solve for \(x\):\[ \frac{2x}{2} = \frac{4}{2} \]This gives:\[ x = 2 \] Therefore, the solution set for the equation \(2x + 4 = 8\) is \( \{2\} \).

Step 3: Simplify Equation 2

Now, simplify the second equation:\[ 2\sin x + 4 = 8 \]Subtract 4 from both sides:\[ 2\sin x + 4 - 4 = 8 - 4 \]This simplifies to:\[ 2\sin x = 4 \]

Step 4: Solve for \(\sin x\) in Equation 2

Proceed by dividing both sides by 2 to solve for \(\sin x\):\[ \frac{2\sin x}{2} = \frac{4}{2} \]This gives:\[ \sin x = 2 \]

Step 5: Analyze the Range of \(\sin x\)

The sine function, \(\sin x\), can only have values in the range \([-1, 1]\). Since \(2\) is outside this range, there is no real number \(x\) for which \(\sin x = 2\). Thus, the solution set is the empty set \(\{ \} \).

Key Concepts

Linear Equations

Linear equations are mathematical statements that represent lines when graphed on a coordinate plane. These equations are expressed in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In a linear equation, the highest power of the variable is 1. This linearity makes solving these equations straightforward.To solve a linear equation, you typically follow these steps:

• Isolate the Variable: Move all terms containing the variable to one side of the equation.
• Simplify: Combine any like terms and simplify the equation to isolate the variable.
• Solve: Perform any necessary operations to solve for the variable.

For example, with the equation \( 2x + 4 = 8 \), you start by moving the constant term to the other side, resulting in \( 2x = 4 \). Then, divide both sides by the coefficient of \( x \), which gives \( x = 2 \).
This solution process makes linear equations a key concept of algebra.

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. These equations can be more complex to solve because they deal with angles and periodic functions. A simple example of a trigonometric equation is \( 2 \sin x + 4 = 8 \).The steps to solve a trigonometric equation are as follows:

• Simplify the Equation: Rearrange the equation to extract the trigonometric function.
• Isolate the Trigonometric Function: Solve for the function by performing algebraic operations.
• Analyze Function Range: Check if the isolated value is within the valid range of the trig function.

For example, in the solution, \( \sin x = 2 \), it's critical to realize that the sine function only ranges from \(-1\) to \(1\). Since the solution here is beyond this range, no real angle satisfies this equation.
This is why understanding the nature of trigonometric functions is crucial in solving these equations.

Solution Sets

A solution set is the collection of values that satisfy a given equation. It can contain one, multiple, or no solutions, depending on the equation's nature.

• Single Solution: Some equations, like many linear equations, result in a single value solution. For instance, the solution to \( 2x + 4 = 8 \) is \( \{2\} \).
• Multiple Solutions: Trigonometric equations often have multiple solutions, especially if they involve periodic functions, where angles can repeat at different cycles.
• Empty Set: This occurs when no real solution satisfies the equation, as seen in the trigonometric equation \( 2 \sin x = 4 \). Here, the solution is \( \{ \} \), indicating no possible real-world angle satisfies this version of the equation.

Understanding solution sets is essential for interpreting and solving equations correctly. Recognizing whether a set is empty or contains several solutions can deeply affect the conclusions drawn from the mathematical problem being addressed.