Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos \frac{1}{2}\)

Step-by-Step Solution

Answer

The exact value of \( y = \arccos \frac{1}{2} \) is \( \frac{\pi}{3} \).

Step 1: Understand the ArcCosine Function

The arc cosine function, denoted as \( \arccos \), is the inverse of the cosine function, and it returns the angle whose cosine is the specified value. It's important to recognize that the range of \( \arccos \) is \([0, \pi]\), which corresponds to angles on the unit circle.

Step 2: Recognize the Trigonometric Identity

We need to find the angle \( y \) such that \( \cos(y) = \frac{1}{2} \). Within the range \([0, \pi]\), the angles where this is true are standard angles in the unit circle.

Step 3: Identify the Standard Angle in Radians

From trigonometry, we know that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). Therefore, in this context of \( \arccos \), the angle \( y \) which satisfies \( \cos(y) = \frac{1}{2} \) in the range \([0, \pi]\) is \( \frac{\pi}{3} \).

Step 4: Write the Final Answer

Thus, the exact value of \( y \) in radians, where \( y = \arccos \frac{1}{2} \), is \( \frac{\pi}{3} \).

Key Concepts

Cosine Function

The cosine function, often written as \( \cos(\theta) \), is one of the primary trigonometric functions. It relates the angle \( \theta \) of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In the context of the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. This makes the cosine function periodic, with a period of \( 2\pi \).
In mathematical terms, the cosine function is defined for all real numbers, and it takes values between -1 and 1, inclusive. This range is important as it defines the possible output values for the cosine function when any real number is used as input. The typical waveform of the cosine function looks like a wave oscillating between these maximum and minimum values. Understanding this helps in comprehending how the inverse cosine function, like \( \arccos \), works.

Unit Circle

The unit circle is a crucial concept in trigonometry and provides a geometric perspective to understand trigonometric functions like cosine. It is a circle with a radius of one, centered at the origin \( (0, 0) \) in the coordinate plane. Every angle \( \theta \) on the unit circle corresponds to a point \((x, y)\), where \( x = \cos(\theta) \) and \( y = \sin(\theta) \).

The significance of the unit circle lies in its ability to map angles directly to values of trigonometric functions. Each point on the circumference represents the endpoint of a vector having the same angle from the positive x-axis. For instance, for \( y = \arccos(\frac{1}{2}) \), you find the point where \( x = \frac{1}{2} \), which corresponds to the angle \( \frac{\pi}{3} \) because at this angle, the x-coordinate matches \( \frac{1}{2} \). This visualization helps in easily determining the trigonometric values and their inverse values.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of the basic trigonometric functions. For example, while the cosine function takes an angle and returns a ratio, the arccosine function does the opposite. It takes a ratio and returns an angle. These functions are very useful in situations where you need to determine the angle that corresponds to a specific trigonometric ratio.

The general function of inverse trigonometry is to allow us to work backwards from the ratio to find the angle. The output of the \( \arccos \) function is constrained between \([0, \pi]\), which means it can only return angles within this range, suitable for many practical applications such as physics and engineering. For instance, if \( \cos(y) = \frac{1}{2} \), then \( y \) within \([0, \pi]\) is \( \frac{\pi}{3} \) because \( \arccos(\frac{1}{2}) \) provides the angle whose cosine is \( \frac{1}{2} \), evidently linking practical problems to their solutions.