Problem 28 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arccos \left(\cos \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

\( \frac{2\pi}{3} \)

Step 1: Understand the Problem

We need to find the exact value of the expression \( \arccos \left(\cos \frac{2\pi}{3}\right) \). This implies finding the angle whose cosine is equal to \( \cos \frac{2\pi}{3} \).

Step 2: Evaluate the Cosine

The angle \( \frac{2\pi}{3} \) is in the second quadrant. In this quadrant, the cosine of an angle is negative. Thus, \( \cos \frac{2\pi}{3} = -\cos \left( \pi - \frac{2\pi}{3} \right) \). So, \( \cos \frac{2\pi}{3} = -\cos \left( \frac{\pi}{3} \right) = -\frac{1}{2} \).

Step 3: Solve the Arccos Function

The \( \arccos \) function returns the angle whose cosine is the given value, usually in the range \([0, \pi] \). So, we are looking for an angle \( \theta \) such that \( \cos \theta = -\frac{1}{2} \).

Step 4: Identify the Angle

The angle \( \theta \) that satisfies \( \cos \theta = -\frac{1}{2} \) within \([0, \pi] \) is \( \theta = \frac{2\pi}{3} \). This is simply because we started with the cosine of this angle.

Step 5: Conclusion

Therefore, the exact value of \( \arccos \left( \cos \frac{2\pi}{3} \right) \) is \( \frac{2\pi}{3} \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates the angle to the horizontal coordinate of a point on the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane.

• The cosine of an angle is the x-coordinate of the corresponding point on the unit circle.
• It is an even function, which means that \( \cos(-x) = \cos(x) \).
• Its range is from \(-1\) to \(1\), inclusive.

For example, in the exercise, we evaluated \( \cos \frac{2\pi}{3} \), which lies in the second quadrant of the unit circle where the cosine values are negative. This results in \( \cos \frac{2\pi}{3} = -\frac{1}{2} \). Understanding the behavior of the cosine function in different quadrants is crucial for solving trigonometric expressions and evaluating them precisely.

Arccosine Function

The arccosine function, denoted as \( \arccos(x) \), is the inverse of the cosine function. When we say inverse, it means finding the angle whose cosine is \( x \). The result is an angle in the range of \([0, \pi] \).

• The output of \( \arccos \) is the angle that when used in the cosine function gives the input value.
• It is useful for determining angles from known cosine values.

In our current exercise, \( \arccos(-\frac{1}{2}) \) was evaluated to find the angle \( \theta \) such that \( \cos \theta = -\frac{1}{2} \).Knowing the exact value of this function is essential for solving exercises without relying on calculators. The recognized range helps to pinpoint the correct angle, ensuring that the solution is precise and fits within the standard interval.

Unit Circle

The unit circle is a powerful tool in trigonometry. It simplifies understanding how trigonometric functions work and relate to angles. Here’s why it’s essential:

• The circle's radius is 1, which simplifies calculations.
• Points on this circle can represent both the sine and cosine of angles.
• It helps in memorizing the sine and cosine values for common angles.

On the unit circle, the angle \( \frac{2\pi}{3} \) is positioned in the second quadrant, where only the sine values are positive. Hence, cosine \( \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \).Visualizing angles and their trigonometric values on the unit circle is important for understanding their behavior and relationships. It also assists in retrieving exact trigonometric values effortlessly.

Exact Values of Trigonometric Functions

Recognizing exact values of trigonometric functions is crucial for evaluating expressions and solving equations. These are key points you should keep in mind:

• Common angles such as \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \) have known sine and cosine values.
• It's valuable to memorize these values to quickly solve expressions like \( \arccos(\cos \frac{2\pi}{3}) \).
• These precise values are particularly helpful in calculus and higher-level mathematics.

In the exercise, we determined that the exact value of \( \arccos(-\frac{1}{2}) \) is \( \frac{2\pi}{3} \). This understanding relies on knowing that \( \cos \) was originally \(-\frac{1}{2} \)of this angle.Mastering these exact values allows for more effective problem-solving, giving you confidence in pinpointing accurate solutions without electronic help.