Problem 30 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cos \left[\operatorname{Arccos}\left(-\frac{1}{2}\right)\right]\)

Step-by-Step Solution

Answer

\(-\frac{1}{2}\)

Step 1: Understand the Problem

The problem wants us to find the cosine of the angle whose cosine value is \(-\frac{1}{2}\). This involves understanding the inverse function \(\operatorname{Arccos}\).

Step 2: Identify the Function's Range

The inverse cosine function, \(\operatorname{Arccos}\), gives us an angle between 0 and \(\pi\) radians where the cosine of that angle equals the input value. So, \(\operatorname{Arccos}\left(-\frac{1}{2}\right)\) will result in an angle in that range.

Step 3: Determine the Angle

Knowing that \(\cos(\pi/3) = \frac{1}{2}\) and due to cosine's symmetries, \(\cos(\pi - \pi/3) = -\frac{1}{2}\), the angle that satisfies \(\cos(\text{angle}) = -\frac{1}{2}\) is \(\frac{2\pi}{3}\) radians.

Step 4: Apply the Cosine Function

Apply the cosine function on the angle obtained from the inverse cosine: \(\cos\left(\operatorname{Arccos}\left(-\frac{1}{2}\right)\right) = \cos\left(\frac{2\pi}{3}\right)\). This equals \(-\frac{1}{2}\) due to our choice of angle.

Key Concepts

Inverse trigonometric functions

Inverse trigonometric functions help us find angles when we know certain trigonometric values. They are essentially the reverse process of the basic trigonometric functions. For example, if the cosine of an angle is known, the inverse cosine, denoted as \( \operatorname{Arccos} \), will tell us what the angle is.

These functions are pivotal when working with angles in different contexts. Knowing one side of an equation allows us to reverse-engineer the angle that fits. Take \( \operatorname{Arccos} \) as an example. It gives back an angle in the range of \( 0 \) to \( \pi \) radians, which can be useful in various calculations, especially when dealing with parts of a circle or periodic functions in mathematics.

Importantly, since cosine values can be negative or positive, the \( \operatorname{Arccos} \) function specifically targets this range for its result to provide the correct corresponding angle. This specificity is essential to solving problems uniquely and correctly, ensuring that we have consistent and interpretable results.

Radians

Radians are a way of measuring angles using the radius of a circle. Unlike degrees, radians measure angles in terms of \( \pi \). The entire circumference of a circle in radians is \( 2\pi \), which corresponds to \( 360 \) degrees.

Understanding radians is crucial when working with trigonometric functions like cosine and their inverses. The concept of radians creates a more natural and direct way to relate angles to the number of rotations or portions of the circle. For instance, \( \pi \) radians is half a circle, \( \frac{\pi}{3} \) radians represents a 60-degree slice, and similarly, \( \frac{2\pi}{3} \) is two-thirds of \( \pi \), aligning with an angle's position as required by a problem like this one.

Radians make it easier to work mathematically with periodic concepts, especially when every calculation in mathematics needs a consistent system. They ensure when we talk about trigonometric cycles, we can do so precisely and without ambiguity.

Cosine function

The cosine function is one of the primary trigonometric functions, often used to relate the angle of a right-angled triangle to the lengths of the adjacent side and hypotenuse.

In mathematics, the cosine function is crucial not only in geometry but also in periodic phenomena, where it helps describe oscillations and waveforms. The function is part of the cosine wave which mimics repetitive or circular movement patterns.

Cosine can take inputs that are either in the numerical form or the unit circle's angle, and its output ranges from \(-1\) to \(1\). This characteristic means that the function can denote the stretched or compressed cycles of waves or rotations. In the context of our problem, understanding the cosine function helps recognize that \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \), showing the cosine value for an angle within the function's typical use-case.

Apart from applications in geometry, the cosine function is also extensively used in physics, engineering, and other fields where wave and oscillation patterns are analyzed.