Problem 152 - Precalculus Essentials

Question

Determine whether each -statement makes sense or does not make sense, and explain your reasoning. I solved \(4 \cos ^{2} x=5-4 \sin x\) by working independently with the left side, applying a Pythagorean identity, and transforming the left side into \(5-4 \sin x\)

Step-by-Step Solution

Answer

The statement does not make sense due to incorrect transformation of the Pythagorean identity.

Step 1: Analyze the equation

The given equation is \(4 \cos ^{2} x=5-4 \sin x\). The student claims to have transformed \(4 \cos ^{2} x\) to \(5-4 \sin x\) by using a Pythagorean identity.

Step 2: Examine the correctness of the transformation

Pythagorean identities in trigonometry are \(\sin^2 x + \cos^2 x = 1\), \(1 + \tan^2 x = \sec^2 x\), and \(1 + \cot^2 x = \csc^2 x\). When working with the left hand side (LHS) of the given equation, we can see that the first Pythagorean identity can be used to change expressions involving \(\cos^2 x\) into expressions involving \(\sin x\). By rearranging the first identity, we get \(\cos^2 x = 1 - \sin^2 x\) and could plug this back into the LHS of the original equation to form \(4(1-\sin^2 x) = 5 - 4 \sin x\). However, this does not simplify to \(5 - 4 \sin x\). Therefore, the student's work is incorrect.

Step 3: Provide the correct reason

The work of the student does not make sense because the transformation from \(4 \cos ^{2} x\) to \(5-4 \sin x\) using the Pythagorean identity was applied incorrectly.

Key Concepts

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. Solving them involves finding the values of the variables, usually angles, that make the equation true.
These types of equations often appear in many mathematical and real-world applications, from determining the heights of buildings using angles to predicting the positions of planets.
When working with trigonometric equations, you typically:

• Identify known trigonometric identities that can simplify or transform the equation.
• Apply algebraic techniques and trigonometric identities to isolate the trigonometric function.
• Solve the resultant simplified expression, often by using inverse trigonometric functions or other methods.

Remember that some solutions might be extraneous, which means they are not valid within the given domain. Therefore, always check your solutions against the original equation.

Cosine and Sine Relationship

The relationship between cosine (\(\cos x\)) and sine (\(\sin x\)) is essential in trigonometry, especially when using identities to solve equations.
The most fundamental identity connecting cosine and sine is the Pythagorean identity:

• \(\sin^2 x + \cos^2 x = 1\)

To transform \(\cos^2 x\) into a sine expression, you can rearrange the identity to get:

• \(\cos^2 x = 1 - \sin^2 x\)

This transformation is often used to switch between trigonometric functions, making complex problems simpler or more solvable.
Mistakes, such as incorrectly applying transformations like converting \(4 \cos^2 x\) directly to\(5 - 4 \sin x\), disrupt understanding and solution of problems. Being vigilant about correctly applying these identities ensures accurate results.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to find solutions to mathematical problems. It's a crucial skill that underlies all aspects of mathematics.
When solving equations, especially in trigonometry, logical reasoning helps ensure that all steps in a solution are valid and follows a clear, rational path.

• Start by analyzing the problem or equation and understand what is being asked.
• Use known information such as identities and properties to help solve the problem.
• Apply a step-by-step approach, showing why each step is valid and leads to the next.

For example, in the discussed exercise, proper reasoning would involve testing whether transformations applied via identities were correctly derived and logical. Identifying errors through logical reasoning allows you to correct the path and find the correct solution.
Regular practicing of logical methods encourages sharper problem-solving skills and deeper understanding of mathematical principles.