Problem 56 - Algebra 2

Question

Error Analysis A student solved an equation as shown below. What error did the student make? $$\begin{aligned} \theta &=\cos ^{-1} 0.5 \\ &=\frac{1}{\cos 0.5} \\ & \approx \frac{1}{0.88} \\ & \approx 1.14 \end{aligned}$$

Step-by-Step Solution

Answer

The error occurs in the second step, where the student incorrectly interpreted the inverse cosine function (\(\cos^{-1}\)) as the reciprocal of cosine. The correct interpretation of \(\cos^{-1}(0.5)\) is the angle whose cosine is \(0.5\), which is \(60^\circ\) or \(\frac{\pi}{3}\) radians.

Step 1: Identify Incorrect Step

Look at the steps closely. The first step of solving for \(\theta\) involves finding the inverse cosine of \(0.5\). However, in the second step, the student falsely interprets \(\cos^{-1} 0.5\) as \(\frac{1}{\cos 0.5}\). This is incorrect and is where the mistake resides.

Step 2: Understand Inverse Trigonometric Function

\(\cos^{-1}(x)\), also written as \(\arccos(x)\), is not the reciprocal of cosine as the notation might imply. Rather, it refers to the angle whose cosine is \(x\). A key thing to understand here is that for \(\cos x = 0.5\), \(x\) could be 60 degrees or \(\frac{\pi}{3}\) radians (or any multiples), not \(\frac{1}{0.88}\).

Step 3: Corrections to Given Steps

Therefore, the correct way of calculating the inverse cosine is to use the specific mathematical function for it, \(\cos^{-1}\), instead of taking the reciprocal. So, \(\cos^{-1}(0.5)\) gives the angle whose cosine is \(0.5\). Using a calculator, or remembering the unit circle, shows that this angle is \(60^\circ\) or \(\frac{\pi}{3}\) radians.

Key Concepts

Cosine Function

In trigonometry, the cosine function is fundamental. It relates the angle of a right-angled triangle to the ratio of the length of the adjacent side over the hypotenuse. Mathematically, if you have an angle \( \theta \), and the lengths of the sides are known, you can find the cosine using the equation

• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

For example, if a triangle has an angle \( \theta \) and a cosine value of \( 0.5 \), you are essentially saying that the length of the side adjacent to the angle is half the length of the hypotenuse.
When we deal with inverse trigonometric functions, like \( \cos^{-1}(x) \), it's essential to recognize that this function gives the angle whose cosine is \( x \).
Inverse functions undo what the original trigonometric function does. For instance,

• \( \cos(\theta) = 0.5 \rightarrow \theta = \cos^{-1}(0.5) \)
• The solution here would be \( \theta = 60^\circ \) or \( \frac{\pi}{3} \) radians.

Ultimately, the cosine function is a core building block in understanding trigonometry and serves as the basis of many more complex concepts.

Error Analysis

Error analysis is a critical skill in mathematics and problem-solving. When identifying errors, it's important to closely examine each step in a problem-solving process to determine where misconceptions or inappropriate calculations may have occurred.
In the provided exercise, the student made an error by confusing the inverse cosine function with the reciprocal of the cosine function. This was the key mistake that led to an incorrect calculation. Generally, error analysis includes:

• Reviewing every step of a solution to find deviations from standard procedures.
• Understanding the properties and definitions of the mathematical functions involved.
• Correcting these misconceptions by referring back to the correct mathematical principles.

Remember to differentiate between an inverse and a reciprocal function:

• The inverse cosine of \( 0.5 \) gives the angle whose cosine is \( 0.5 \).
• The reciprocal of \( \cos(x) \) would be \( \frac{1}{\cos(x)} \).

These distinctions are fundamental in performing accurate and meaningful mathematical analyses.

Mathematical Notation

Mathematical notation is a system of symbols used to convey concepts and operations in mathematics clearly. Its proper use is vital in communicating ideas accurately. Each symbol or notation has a specific, often universally accepted meaning.
In trigonometry, for example, \( \cos^{-1}(x) \) or \( \arccos(x) \) represents the inverse cosine function. This particular notation means the function returns the angle whose cosine is \( x \). However, this can be mistakenly perceived as a reciprocal operation. Always remember:

• \( \cos^{-1}(x) \) is about finding angles, not fractions.
• Notations and symbols help prevent misinterpretations in complex equations.

Ensuring correct usage is key in avoiding errors. As shown in the original solution, confusing these two notations led to a fundamental error. Thus, mastering mathematical notation is crucial for any student tackling exercises involving algebra, geometry, or trigonometry. It's the language that makes the intricate concepts accessible, maintaining precision across various mathematical problems.