Problem 7 - Advanced Engineering Mathematics

Question

In Problems 1-14, find all values of the given quantity. $$ \cos ^{-1} \frac{1}{2} $$

Step-by-Step Solution

Answer

\( \cos^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{3} \).

Step 1: Understanding the Problem

The problem asks us to find the angle whose cosine value is \( \frac{1}{2} \). This is the same as finding \( \theta \) where \( \cos(\theta) = \frac{1}{2} \). The function \( \cos^{-1} \) represents the inverse cosine or arccosine function.

Step 2: Identify the Principal Value Range

The range of the \( \cos^{-1} \) function is from 0 to \( \pi \) radians. This is the principal value range for arccosine. We need to find \( \theta \) such that \( 0 \leq \theta \leq \pi \).

Step 3: Finding the Solution from Unit Circle

On the unit circle, \( \cos(\theta) = \frac{1}{2} \) occurs at \( \theta = \frac{\pi}{3} \) radians. This is the angle between 0 and \( \pi \) where cosine gives \( \frac{1}{2} \).

Step 4: Conclusion

Since \( \frac{\pi}{3} \) is within the range [0, \( \pi \)], it is the value returned by \( \cos^{-1}(\frac{1}{2}) \).

Key Concepts

Cosine Function

The cosine function, often abbreviated as "cos," is one of the fundamental trigonometric functions. It measures the horizontal coordinate of an angle on the unit circle. This function takes an angle as an input and provides the cosine value of that angle as an output.
Understanding the cosine function is crucial for solving trigonometric problems, especially those involving inverse trigonometric functions like arccos.
Some key points about the cosine function include:

• The cosine of an angle in the unit circle is the x-coordinate of the endpoint of the arc made by that angle.
• The cosine function is periodic, meaning it repeats its values in a regular pattern at specific intervals.
• The range of the cosine function is from -1 to 1. This range represents the maximum and minimum x-values possible on the unit circle for any given angle.

In contexts like this exercise, understanding the cosine value allows you to find angles when given specific cosine values, using inverse functions like \( \cos^{-1} \).

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a critical tool in trigonometry because it provides a simple and clear way to define trigonometric functions for all angles.
Here's why the unit circle is essential:

• The unit circle allows us to visualize the angles and their corresponding trigonometric values (sine, cosine, and tangent).
• It provides a geometric interpretation: cosine represents the x-coordinate, while sine represents the y-coordinate of a point on the circle.
• The angles on a unit circle are usually given in radians, which are a different measure than degrees but equally important in advanced mathematics.

To find when \( \cos(\theta) = \frac{1}{2} \), as in this exercise, we look for points on the unit circle where the x-coordinate is \( \frac{1}{2} \). From what we know, these points are at angles \( \frac{\pi}{3} \) and \( \frac{5\pi}{3} \). However, only \( \frac{\pi}{3} \) lies within the principal value range of arccos.

Principal Value

The concept of the principal value is crucial when dealing with inverse trigonometric functions, like inverse cosine. It refers to the specific range of angle values that an inverse function returns. This range ensures that each output remains unique and does not repeat, which is important for the function to be well-defined.
For the inverse cosine function, \( \cos^{-1} \), the principal value range is from 0 to \( \pi \) radians.
Understanding this concept helps in solving problems where multiple solutions might exist mathematically, but you're asked for a single "principal" or primary solution.
Here's how the principal value applies to this exercise:

• The range \(0 \leq \theta \leq \pi\) ensures that there is a unique value for \( \cos(\theta) = \frac{1}{2} \), which is \( \theta = \frac{\pi}{3} \), as other solutions like \( \theta = \frac{5\pi}{3} \) lie outside this range.
• The concept of principal value helps avoid ambiguity by setting a standard interval to find these inverse values, providing consistency in multi-solution context.