Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\cos ^{-1}\left(-\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Inverse Cosine Definition

The function \(cos^{-1}(x)\) or arccos(x) represents the angle \(y\) whose cosine is \(x\). It is defined for inputs between [-1,1], and the output is an angle in the range \([0, \pi]\).

Step 2: Recognize the Known Value of Cosine

The cosine of an angle \(y\) is \(-\frac{1}{2}\). We know from trigonometric tables that \(\cos(\frac{2\pi}{3}) = -\frac{1}{2}\).

Step 3: Identify the Range of the Angle

The range of the arccos function is \([0, \pi]\). The angle \(\frac{2\pi}{3}\) lies within this interval.

Step 4: Solve for \(y\)

Since \(\cos(y) = -\frac{1}{2}\), and the angle must be within the range of \(\cos^{-1}\), we find that \(y = \frac{2\pi}{3}\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry. It describes the relationship between the angle of a right triangle and the ratio of the adjacent side to the hypotenuse. Understanding it helps us form a solid basis for analyzing oscillations, circles, and waves.

When we say "cosine," we often denote it as \( \cos(\theta) \). It assigns an angle \( \theta \) to a value between \(-1\) and \(1\).

• Cosine is periodic with a period of \(2\pi\).
• It is an even function, meaning \(\cos(\theta) = \cos(-\theta)\).
• The cosine of 0 is 1, and as \(\theta\) progresses to \(\pi\), the cosine decreases to -1.

These properties are vital when solving problems involving angles and lengths, such as determining where the cosine of a particular angle falls on the unit circle.

Arccos

Arccos, or inverse cosine, is the function that tells us what angle corresponds to a given cosine value. It acts as the reverse operation of the cosine function, converting a value back to an angle. This function is denoted as \( \cos^{-1}(x) \).

The output of arccos is an angle in radians, specifically between 0 and \(\pi\):

• It is restricted to this range by definition to maintain it as a function (one output for each input).
• When \(x = 1\), \(\cos^{-1}(x) = 0\).
• When \(x = 0\), \(\cos^{-1}(x) = \frac{\pi}{2}\).
• And when \(x = -1\), \(\cos^{-1}(x) = \pi\).

Using arccos helps in solving trigonometric equations and is crucial in situations where the cosine value of an angle is known but not the angle itself.

Trigonometric Tables

Trigonometric tables are essential tools, especially when dealing with exact values of trigonometric functions without a calculator. These tables include common angles such as 0, \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \) and more.

To determine the cosine of these angles and others, the tables help us easily refer to:

• Information about sine, cosine, and tangent values.
• Specific angles known for simple fractions, like \(\frac{1}{2}, \sqrt{2}/2, \sqrt{3}/2,\) etc.

By utilizing these known values, students can deduce the cosine of angles even if the angle isn't common, such as \(\frac{2\pi}{3}\). These tables provide a shortcut to understanding the inverse trigonometric functions by allowing rapid cross-reference of angles and their cosines.

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin on a coordinate plane. It represents the fundamental bridge between algebraic and geometric perspectives of the trigonometric functions.

Every angle on the unit circle corresponds to a point \((x, y)\), where:

• \(x\) is the cosine of the angle.
• \(y\) is the sine of the angle.

Understanding the unit circle allows us to easily see how cosine and sine functions change over different quadrants:

• In the first quadrant, \(x\) is positive, aligning with positive cosine values.
• As you move to the second quadrant, cosine becomes negative, illustrating the \(\cos(y)=-\frac{1}{2}\) result.

By using the unit circle, you can visually grasp how the cosine function acts and reflects symmetry. It also helps determine special angles without exact calculation, showing how trigonometric functions mirror and repeat.