Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\arccos \left(-\frac{\sqrt{3}}{2}\right)$$

Step-by-Step Solution

Answer

\(y = \frac{5\pi}{6}\)

Step 1: Understand the Problem

We need to find the angle \(y\) whose cosine value is \(-\frac{\sqrt{3}}{2}\). This is expressed as \(y = \arccos\left(-\frac{\sqrt{3}}{2}\right)\). The arccos function returns angles in the range \([0, \pi]\) radians.

Step 2: Recall the Unit Circle Properties

On the unit circle, an angle whose cosine value is \(-\frac{\sqrt{3}}{2}\) corresponds to the standard position angle where the x-coordinate is \(-\frac{\sqrt{3}}{2}\). This occurs at \(150^\circ\) and \(210^\circ\). In radians, these angles are \(\frac{5\pi}{6}\) and \(\frac{7\pi}{6}\).

Step 3: Choose the Correct Angle

Since \(\arccos\) returns an angle in the range \([0, \pi]\), \(\frac{5\pi}{6}\) is within this range, but \(\frac{7\pi}{6}\) is not (since \(\frac{7\pi}{6} > \pi\)). Therefore, the correct angle is \(\frac{5\pi}{6}\).

Step 4: Conclude the Solution

Thus, the exact value of \(y = \arccos \left(-\frac{\sqrt{3}}{2}\right)\) is \(\frac{5\pi}{6}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of one, centered at the origin of the coordinate plane. This simple setup allows us to easily relate angles and trigonometric functions.

• The unit circle is defined by the equation: \[ x^2 + y^2 = 1 \]
• Angles on the unit circle are measured from the positive x-axis. They go counterclockwise for positive angles and clockwise for negative angles.
• The coordinates of a point on the unit circle can be related to trigonometric functions: \( (x, y) = (\cos(\theta), \sin(\theta)) \).

For instance, when using the unit circle to understand inverse trigonometric functions, like arccosine, it helps to visualize the x-coordinate, which represents the cosine value we're interested in. For our problem, this x-coordinate is \(-\frac{\sqrt{3}}{2}\). Thus, we locate where on the unit circle this value occurs.
Knowing standard angles and their coordinates on the unit circle is essential in solving problems without a calculator. Each angle in both degrees and radians corresponds to a point on the unit circle, making it easier to identify possible values for trigonometric functions.

Radians

Radians are a way of measuring angles based on the radius of a circle. This unit of measurement is widely used in mathematics because it provides a natural way to express angles in terms of the circle's geometry.

• One full rotation around a circle is \(2\pi\) radians, which is equivalent to 360 degrees.
• Therefore, \(\pi\) radians equals 180 degrees, and \(\frac{\pi}{3}\) radians equals 60 degrees, as examples.
• Radian measure is defined such that the length of the arc subtended by the angle is equal to the radius of the circle.

Using radians is particularly useful when we work with trigonometric functions, calculus, and periodic functions because it simplifies mathematical expressions. In our initial exercise, we directly translated the degrees to radians to find the solution within the appropriate range for arccosine.
I've highlighted that the solution used the range \([0, \pi]\) radians for the arccos function. Here, an angle of \(\frac{5\pi}{6}\) radians was selected because it falls perfectly within this range, whereas \(\frac{7\pi}{6}\) radians falls outside the valid range.

Cosine Function

The cosine function is a fundamental trigonometric function that relates an angle to the ratio of the adjacent side over the hypotenuse in a right triangle.

• In the context of the unit circle, it represents the x-coordinate of a point on the circle.
• The cosine function is periodic, with a period of \(2\pi\), meaning that every \(2\pi\) radians, the function repeats its values.
• Cosine values range from \(-1\) to \(1\), as these represent the maximum leftward and rightward boundaries on the unit circle's x-axis.

Understanding the cosine function is crucial when working with inverse trigonometric functions. Specifically, with \(\arccos\), we're determining which angle within the function's range corresponds to a given cosine value. Here, since \(-\frac{\sqrt{3}}{2}\) falls within the possible output of cosine, we use this information to identify potential angles on the unit circle.
The solution aimed to find where the angle corresponds to this cosine value, specifically using the inverse cosine, which has a range of \([0, \pi]\) radians. Ultimately, \(\frac{5\pi}{6}\) was the correct angle within this range, providing the exact value of \(y\).