Problem 34 - Precalculus Essentials

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The exact value is \(\frac{2\pi}{3}\).

Step 1: Find the cosine of the given angle

Since the cosine function is periodic with period \(2\pi\), the value of \(\cos(\frac{2\pi}{3})\) is the same as \(\cos(\frac{2\pi}{3} - 2\pi)\), as subtracting any multiple of \(2\pi\) from the angle does not change the value of the cosine. Therefore, \(\cos(\frac{2\pi}{3}) = \cos(\frac{2\pi}{3} - 2\pi) = \cos(\frac{2\pi}{3} - 2\pi) = \cos(-\frac{4\pi}{3}) = -\frac{1}{2}\).

Step 2: Apply the inverse cosine

The next step is to apply the \(\cos^{-1}\) to the output from Step 1. The principal value of \(\cos^{-1}(-\frac{1}{2})\) is \(\frac{2\pi}{3}\), since \(\cos(\frac{2\pi}{3}) = -\frac{1}{2}\). That is because the range of \(\cos^{-1}(x)\) is [0, \(\pi\)], which includes \(\frac{2\pi}{3}\).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(\theta) \), is one of the primary trigonometric functions. It is used to describe the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• Cosine values range between -1 and 1 inclusive.
• It is periodic with a period of \( 2\pi \), meaning that its values repeat every \( 2\pi \) radians.
• The function is even, which means \( \cos(-\theta) = \cos(\theta) \).

Understanding its periodic nature is crucial, particularly because it enables us to find equivalent cosine values for angles beyond a single rotation of the unit circle. For instance, \( \cos\left(\frac{2\pi}{3}\right) \) reflects the same value as \( \cos\left(-\frac{4\pi}{3}\right) \), showcasing its repetitive sine wave behavior over the mathematical plane.

Principal Value

The principal value is an important concept when dealing with inverse trigonometric functions like \( \cos^{-1} \). It determines the unique angle that a given trigonometric function returns.

• For the cosine inverse function, \( \cos^{-1}(x) \), the principal value range is between 0 and \( \pi \) radians.
• This means that for any \( x \) within the domain -1 to 1, \( \cos^{-1}(x) \) will yield an angle in this range.
• This helps eliminate the ambiguity of angles which are co-terminal, i.e., differing by multiples of \( 2\pi \).

In the case of \( \cos^{-1}\left(\cos\frac{2\pi}{3}\right) \), the principal value is \( \frac{2\pi}{3} \) because it falls naturally within the range of the inverse cosine's principal value domain.

Trigonometric Identities

Trigonometric identities are equations that relate related trigonometric functions to one another, and they are fundamental in simplifying and solving trigonometric problems.

• The cosine of a negative angle identity: \( \cos(-\theta) = \cos(\theta) \).
• Periodic identity for cosine: \( \cos(\theta + 2\pi k) = \cos(\theta) \), where \( k \) is any integer.
• Given the identity, \( \cos(\theta) = \cos(\pi - \theta) \), you can understand how to solve equations involving the cosine function.

By applying these identities, we can transform and solve expressions such as \( \cos^{-1}\left(\cos\frac{2\pi}{3}\right) \) by understanding equivalences and symmetry, reducing complex angles to easier, more manageable ones.