Problem 34 - Contemporary Precalculus

Question

Find the exact functional value without using a calculator. $$\cos ^{-1}(\cos 5 \pi / 4)$$

Step-by-Step Solution

Answer

Answer: \(\frac{\pi}{4}\)

Step 1: Express \(5\pi/4\) in the range of the cosine function

First, we need to express the angle $$5\pi/4$$ in the range where the cosine function is defined. The cosine function has a period of \(2\pi\), thus we can find an equivalent angle in the range \([0, 2\pi]\) as follows: $$5\pi/4 = 2\pi - \pi/4$$ Now, \(2\pi - \pi/4\) is in the range \([0, 2\pi]\), so we can use it in the expression.

Step 2: Evaluate the cosine function

Next, we need to evaluate the cosine function for the given angle \(2\pi - \pi/4\). Since the cosine function is an even function, we have: $$\cos(2\pi - \pi/4) = \cos(\pi/4)$$ We know that \(\cos(\pi/4) = \frac{1}{\sqrt{2}}\).

Step 3: Evaluate the inverse cosine function

Finally, we will evaluate the inverse cosine function for the value \(\frac{1}{\sqrt{2}}\). Since \(\cos(\pi/4) = \frac{1}{\sqrt{2}}\), the exact functional value of the given expression is: $$\cos^{-1}(\cos(5\pi/4)) = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \pi/4$$

Key Concepts

Cosine Function

The cosine function, \(\cos(\theta)\), is one of the primary trigonometric functions. It is fundamentally related to the coordinates of a point on the unit circle. In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse. \[\cos(\theta) = \frac{{\text{adjacent}}}{{\text{hypotenuse}}}\]The cosine function is periodic with a period of \(2\pi\), meaning that \(\cos(\theta) = \cos(\theta + 2\pi n)\) for any integer \(n\). This periodicity allows us to find equivalent angles by adding or subtracting \(2\pi\) multiples.

• The range of the cosine function is \([-1, 1]\).
• It is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• It reaches a maximum value of 1 at \(\theta = 0, 2\pi, 4\pi, \ldots\).
• It reaches a minimum value of -1 at \(\theta = \pi, 3\pi, 5\pi, \ldots\).

Understanding these properties helps in solving trigonometric problems effectively, such as finding functional values over specified intervals without the use of a calculator.

Arccosine Function

The arccosine function, often represented as \(\cos^{-1}(x)\), is the inverse of the cosine function. It is used to determine the angle \(\theta\) when the cosine value is known. Because the cosine function is not one-to-one over all real numbers, its inverse is restricted to a specific range: from \(0\) to \(\pi\).

• For any \(x\) in the range \([-1, 1]\), \(\cos^{-1}(x)\) gives the angle between \(0\) and \(\pi\) whose cosine is \(x\).
• It helps resolve trigonometric equations where the angle must be found for a known cosine value.

When solving expressions like \(\cos^{-1}(\cos(\theta))\), the challenge is to express \(\theta\) within the valid domain of the inverse function. This often involves adjusting \(\theta\) to fall within \([0, \pi]\), leading to a correct solution.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. These are essential tools for simplifying expressions and proving other mathematical formulas. One of the most significant identities is the Pythagorean identity: \[\sin^2(\theta) + \cos^2(\theta) = 1\]This identity is derived from the Pythagorean theorem and is fundamental in any trigonometric solution. Here are a few important types of identities:

• Even-Odd Identities: \(\cos(-\theta) = \cos(\theta)\) and \(\sin(-\theta) = -\sin(\theta)\).
• Angle Sum and Difference Identities: Useful for finding the trigonometric function of sum or difference of two angles, like \(\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\).

These identities can be crucial when working through problems like reducing angles or expressing functions within specific ranges, as seen in the solution for the problem \(\cos^{-1}(\cos(5\pi/4))\), where even-odd identities and the symmetry of the cosine function were utilized to simplify the expression.