Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arccos \frac{1}{2}\) (b) arccos 0

Step-by-Step Solution

Answer

(a) \(\arccos \left(\frac{1}{2}\right) = 60^{\circ}\) or \(\frac{\pi}{3}\) radians, (b) \(\arccos 0 = 90^{\circ}\) or \(\frac{\pi}{2}\) radians.

Step 1: Solve for \(\arccos \frac{1}{2}\)

Recall that \(\arccos(x)\) is the angle whose cosine is \(x\). Therefore, we need to find the angle such that \(\cos(\theta) = \frac{1}{2}\). From the unit circle, we know that \(\cos(60^{\circ}) = \frac{1}{2}\), or in radians, \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\). Thus, \(\arccos \left(\frac{1}{2}\right) = 60^{\circ}\) or \(\frac{\pi}{3}\) radians. Note that we chose the angle in the first quadrant because by definition, \(\arccos(x)\) returns the angle in \([0, \pi]\).

Step 2: Solve for \(\arccos 0\)

Using similar reasoning as in Step 1, we need to find the angle such that \(\cos(\theta) = 0\). From the unit circle, we know that \(\cos(90^{\circ}) = 0\), or in radians, \(\cos\left(\frac{\pi}{2}\right) = 0\). Thus, \(\arccos 0 = 90^{\circ}\) or \(\frac{\pi}{2}\) radians.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 unit centered at the origin of a coordinate system. On this circle, any point can be defined using a pair of coordinates that correspond to the cosine and sine of an angle. The angle is typically measured from the positive x-axis, moving counter-clockwise.

For example, on the unit circle, the point \( (1, 0) \) corresponds to an angle of \( 0 \) radians, where the cosine value is \( 1 \) and the sine value is \( 0 \). This provides a visual way to understand the values of sine and cosine for different angles. When calculating \( \arccos \frac{1}{2} \) as shown in the textbook exercise, we're looking for the angle where the x-coordinate (cosine) is \( \frac{1}{2} \) and the y-coordinate (sine) is not considered. This angle is found in the first quadrant, where \( \cos(60^\circ) = \frac{1}{2} \) or \( \cos(\frac{\pi}{3}) = \frac{1}{2} \) in radian measure.

Cosine Function

The cosine function is one of the basic trigonometric functions and is related to the x-coordinate of a point on the unit circle. For a given angle \( \theta \) measured in the plane, \( \cos(\theta) \) will give the horizontal distance from the origin to the point on the unit circle intersected by the terminal side of the angle. In simpler terms, it reflects how much 'left' or 'right' from the center of the circle you are when looking at the angle's position.

When you are trying to understand \( \arccos(\frac{1}{2}) \) from the textbook exercise, you're essentially searching for the angle \( \theta \) whose cosine value is \( \frac{1}{2} \) or where the point on the unit circle would be halfway along the horizontal radius. Since the cosine function's values range between -1 and 1, the \( \arccos \) function helps to determine the angle for these given values within the restricted domain.

Radian Measure

Radian measure is a way of expressing angles that represents the length of the arc on the unit circle that is encompassed by the angle. One complete revolution around the circle is \( 2\pi \) radians, which is equivalent to \( 360^\circ \) in degree measure. This means that \( \pi \) radians is equal to \( 180^\circ \) and so on.

Using radians can be more natural in the context of trigonometry and calculus as it relates more directly to the unit circle. When \( \arccos(0) \) is solved, like in the textbook example, the angle in radians is \( \frac{\pi}{2} \) because the arc length from the positive x-axis to the point \( (0, 1) \) on the unit circle's circumference is \( \frac{1}{4} \) of the circle's total circumference, hence \( \frac{1}{4} \) of \( 2\pi \) radians.