Problem 15 - Precalculus: A Concise Course

Question

Evaluate the expression without using a calculator. $$ \arccos \left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

\(\arccos \left(-\frac{1}{2}\right) = 2π/3\).

Step 1: Understanding the problem

Arccosine or inverse cosine, \(\arccos(x)\), is an angle whose cosine is \(x\). The range of \(\arccos(x)\) is \(0 ≤ \arccos(x) ≤ π\). We need to find such angle that its cosine is \(-\frac{1}{2}\).

Step 2: Recall cosine values on the unit circle

On the standard unit circle, there are two main angles (in the first and third quadrants) where the cosine equals \(\frac{1}{2}\): one is \(π/3\) or \(60^{\circ}\) and the second one is \(5π/3\) or \(300^{\circ}\). However, since cosine is negative in the second and third quadrants, the angle we are looking is at the second quadrant.

Step 3: Identify the correct quadrant and angle

Since the cosine value is negative (\(-\frac{1}{2}\)), the angle must lie in the second quadrant. The equivalent angle in the second quadrant for \(π/3\) or \(60^{\circ}\) is \(2π/3\) or \(120^{\circ}\). This is because \(180^{\circ} - 60^{\circ} = 120^{\circ}\) or \(π - π/3 = 2π/3\).

Step 4: Conclusion

Therefore, the angle whose cosine is \(-\frac{1}{2}\) is \(2π/3\) or \(120^{\circ}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when we know the ratio of the sides of a right triangle. In trigonometry, each of the primary trigonometric functions (like sine, cosine, and tangent) has an inverse function that undoes its effect. For example, the inverse cosine function, written as \( \arccos(x) \), tells us the angle whose cosine value is \( x \). This is a particularly useful tool when dealing with geometric problems and circle-related calculations.
An important aspect about these inverse functions is their range. For \( \arccos(x) \), the range is usually restricted from \( 0 \) to \( \pi \) radians to provide a unique angle for each possible cosine value. This range ensures that every input within the range has one output, making \( \arccos(x) \) a proper function. This restriction helps us solve problems involving angles and gives us a standard way to identify them on the unit circle.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. Understanding the unit circle can significantly enhance your skills in evaluating trigonometric functions and solving problems involving angles.
One key feature of the unit circle is that it establishes a connection between angles and linear coordinates.

• Each point on the unit circle can be described using an (x, y) coordinate, where \( x \) is the cosine of the angle, and \( y \) is the sine of the angle.
• Using these coordinates, you can easily recognize and recall common angles and their corresponding trigonometric values.
• For instance, on the unit circle, the angle \( \frac{\pi}{3} \) or \( 60^{\circ} \) corresponds to a cosine value of \( \frac{1}{2} \).

By rotating counter-clockwise, angles in the second quadrant, such as \( 2\frac{\pi}{3} \) or \( 120^{\circ} \), often have negative cosine values, which helps us solve trigonometric equations and find inverse trigonometric functions.

Angle Measurement

Angle measurement is a central concept in trigonometry. Angles can be measured in different units, such as degrees and radians, which helps us describe the size of an angle. This measurement is crucial when identifying specific points on the unit circle or when working through problems that involve trigonometric functions.
There are a few common conversions between degrees and radians that are particularly useful to remember:

• \( 180^{\circ} \) equals \( \pi \) radians
• \( 90^{\circ} \) equals \( \frac{\pi}{2} \) radians
• \( 60^{\circ} \) equals \( \frac{\pi}{3} \) radians

Knowing these conversions helps in solving problems like identifying inverse trigonometric functions, such as \( \arccos(\frac{-1}{2}) \), where we found the angle was \( \frac{2\pi}{3} \) or \( 120^{\circ} \). This dual knowledge of angles in degrees and radians is essential when analyzing problems from different perspectives.