Problem 12 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \arccos \left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

\( \arccos(-\frac{1}{2}) = \frac{2\pi}{3} \).

Step 1: Understand the Inverse Trigonometric Function

The expression \( \arccos(x) \) gives us the angle whose cosine is \( x \). In this exercise, we are looking for an angle \( \theta \) such that \( \cos(\theta) = -\frac{1}{2} \).

Step 2: Identify Quadrants for Cosine

Cosine is positive in the first and fourth quadrants and negative in the second and third quadrants. Since we are seeking an angle where the cosine is negative, we should focus on the second quadrant, as \( \arccos(x) \) traditionally yields angles within \([0, \pi]\).

Step 3: Recall Special Angles and Their Cosines

Recall that \( \cos(\pi/3) = \frac{1}{2} \). Therefore, for \( \cos(\theta) = -\frac{1}{2} \), \( \theta \) should be in the second quadrant which is \( \pi - \frac{\pi}{3} \).

Step 4: Calculate the Exact Angle

To find the exact angle, calculate \( \theta = \pi - \frac{\pi}{3} \). Simplifying,\[\theta = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3}. \]This aligns with our interval \([0, \pi]\) for \( \arccos \).

Step 5: Verify the Result

To ensure accuracy, check that \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \). Considering the cosine in the second quadrant using \( \cos(\pi - \theta) = -\cos(\theta) \), we confirm the calculation is correct.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It measures the ratio of the adjacent side to the hypotenuse in a right triangle. Generally, it is denoted as \( \cos(\theta) \), where \( \theta \) is the angle in question. Unlike other functions, cosine has a periodic nature with a cycle that repeats every \( 2\pi \) radians or 360 degrees.
Cosine values range between -1 and 1. This signifies that for any angle \( \theta \), the cosine value will always lie within this interval. This is crucial when finding an angle for a given cosine value, like \( -\frac{1}{2} \).
When dealing with inverse trigonometric functions, the task is to find the angle \( \theta \) for a known cosine value. This is specifically the work of \( \arccos(x) \), which retrieves such an angle \( \theta \) by reversing the cosine function's operation.

Special Angles

Special angles are particular angles often used in trigonometry due to their easily recognizable cosine, sine, and tangent values. Knowing these can significantly simplify solving trigonometric problems without the need for a calculator.
Some of these angles include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \pi \). Good examples include knowing that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \) and \( \cos(\pi - \frac{\pi}{3}) = -\frac{1}{2} \).
These known values help find cosine and sine values of each special angle within different quadrants through simple adjustments. These adjustments involve knowing in which direction and how much to shift the angle.

Second Quadrant

The second quadrant in trigonometry is crucial for understanding angles where cosine is negative but sine is positive. It lies between angles \( \frac{\pi}{2} \) and \( \pi \) (90° to 180°), making it essential for calculations involving wholesale reversals of cosine values.
In simple terms:

• Cosine: Negative
• Sine: Positive

When evaluating \( \arccos(x) \), which typically provides principal values between 0 and \( \pi \), the second quadrant values are essential to identify whenever \( x \) is negative, like \( -\frac{1}{2} \).
Therefore, identifying that \( \frac{2\pi}{3} \) lies in the second quadrant can be immediately useful, knowing \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \). This value aligns perfectly when solving exercises involving this specific interval.

Exact Trigonometric Values

Finding exact trigonometric values is fundamental in simplifying and solving trigonometric expressions without relying on a calculator. It involves recognizing specific trigonometric values that are inherently known, rather than computed through approximation.
For example, if we already know that \( \cos(\frac{\pi}{3}) = \frac{1}{2} \), mirrors in other quadrants maintain this ratio with only sign changes. Particularly in the second quadrant, this same angle becomes \( \pi - \frac{\pi}{3} \), yielding \( \frac{2\pi}{3} \) with a cosine of \(-\frac{1}{2}\).
These manipulations with exact values apply to any trigonometric functions, streamlining problem-solving and making the calculations intuitive. Whether in exams or deeper studies, mastering these values will give a student confident command over trigonometric functions.