Problem 5 - Calculus Single Variable

Question

Calculate the value of the given inverse trigonometric function at the given point. $$ \arccos (1 / 2) $$

Step-by-Step Solution

Answer

The angle \( \theta \) such that \( \cos(\theta) = \frac{1}{2} \) is \( \frac{\pi}{3} \).

Step 1: Understand the Inverse Trigonometric Function

The function \( \arccos \) or arc cosine gives the angle whose cosine is a given number. Here, we need to find the angle whose cosine equals \( \frac{1}{2} \).

Step 2: Recall the Range of Cosine Function

The cosine function, \( \cos(\theta) \), maps angles \( \theta \) from 0 to \( \pi \) to values between -1 and 1. For arc functions, we consider this principal range because it provides a unique solution.

Step 3: Identify Common Angles

Recall the common angle values: \( \cos(0) = 1 \), \( \cos(\frac{\pi}{3}) = \frac{1}{2} \), \( \cos(\frac{\pi}{2}) = 0 \). Since we are asked for \( \cos^{-1}(\frac{1}{2}) \), this corresponds to the angle \( \frac{\pi}{3} \).

Step 4: State the Solution

Based on the common values, the angle \( \theta \) that satisfies \( \cos(\theta) = \frac{1}{2} \) is \( \frac{\pi}{3} \) radians, which is approximately 60 degrees.

Key Concepts

Arc Cosine

Understanding the concept of arc cosine is essential for tackling inverse trigonometric functions. The arc cosine, written as \( \arccos(x) \), is the inverse of the cosine function. It tells us the angle whose cosine value is \( x \). For instance, if you say \( \arccos(\frac{1}{2}) \), you're asking what angle will give you a cosine value of \( \frac{1}{2} \). The arc cosine function has a range of 0 to \( \pi \) radians, which corresponds to 0 to 180 degrees.
This restriction in its range ensures every output corresponds to exactly one angle and helps maintain a one-to-one relationship. Thus, when we compute \( \arccos(\frac{1}{2}) \), we are looking for a single, unique angle within this range.

Common Angle Values

Common angle values in trigonometry are crucial for quickly identifying solutions to problems involving trigonometric functions. These are values that frequently appear and are typically memorized to make calculations straightforward.
For cosine, some of these common angles, along with their cosine values, include:

• \( \cos(0) = 1 \)
• \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \)
• \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \cos(\frac{\pi}{3}) = \frac{1}{2} \)
• \( \cos(\frac{\pi}{2}) = 0 \)

These angles help us solve problems efficiently. They are particularly useful when dealing with inverse trigonometric functions because once we identify a cosine value, we can usually match it to one of these angles quickly. In the case of \( \arccos(\frac{1}{2}) \), we matched it to \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).

Radian Measure

When dealing with trigonometry, especially in inverse functions, the radian measure becomes very important. A radian is another way to measure angles, and it is based on the radius of a circle. A full circle is \( 2\pi \) radians, which is equivalent to 360 degrees.
Understanding this conversion is helpful:

• \( \pi \) radians = 180 degrees
• \( \frac{\pi}{2} \) radians = 90 degrees
• \( \frac{\pi}{3} \) radians = 60 degrees
• \( \frac{\pi}{6} \) radians = 30 degrees

Knowing radians is vital since many trigonometric functions, including arc cosine, often provide outputs in radians. In the solution \( \arccos(\frac{1}{2}) = \frac{\pi}{3} \), it's useful to know this angle is also 60 degrees for more intuitive, everyday understanding.