Problem 64 - Precalculus : Building Concepts and Connections

Question

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\cos \theta=\frac{1}{2}$$

Step-by-Step Solution

Answer

\(\theta\) equals \(60^\circ\)

Step 1: Identify the Given

Identify the given cosine value for \(\theta\), which is \(\cos \theta = \frac{1}{2}\)

Step 2: Use the Inverse Cosine Function

To determine the measure of the angle, utilize the inverse cosine (or 'arccos') function. Apply it as follows: \(\theta = \arccos(\frac{1}{2})\)

Step 3: Calculate Angle Measure

Use a calculator to find the arccos of \(\frac{1}{2}\). Ensure the calculator is set to the correct units (typically degrees). This will yield the result.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is usually abbreviated as 'cos'. It relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. The cosine of an angle is often used in problems involving right triangles and circular motion.
For any angle \( \theta \), the cosine function is defined as:\[\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\]In the exercise provided, you are given a specific cosine value, \( \cos \theta = \frac{1}{2} \). This means that the adjacent side is half the length of the hypotenuse when \( \theta \) is an acute angle. This relationship is critical in solving the problem because it allows us to identify \( \theta \) using inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the measure of an angle when the trigonometric ratios are known. They are the 'reverse' of the regular trigonometric functions. The common inverse trigonometric functions are arcsin, arccos, and arctan, which correspond to sine, cosine, and tangent respectively.
In this scenario, the inverse cosine function, denoted as \( \arccos \), is used. The inverse cosine, \( \arccos(x) \), will give the angle \( \theta \) for which the cosine is \( x \). This is particularly useful for finding angles when the cosine value is known.
The formula for finding \( \theta \) is then:\[\theta = \arccos\left(\frac{1}{2}\right)\]Using a calculator, you can compute \( \arccos\left(\frac{1}{2}\right) \), ensuring that the calculator is set to use degrees if required. This approach provides the solution to the problem by calculating the angle itself.

Acute Angles

An acute angle is any angle less than 90 degrees. These angles are a major focus of right triangle trigonometry because trigonometric ratios like sine, cosine, and tangent are initially defined for acute angles. They provide a simple basis for understanding how these ratios operate within the unit circle and are essential in various geometric and real-world applications.
In the given problem, you are tasked with finding an acute angle \( \theta \) when \( \cos \theta = \frac{1}{2} \). An important property of the cosine function is that it is positive in the first quadrant of the unit circle, where all angles are acute. Consequently, when using the inverse cosine function to find an angle, it inherently gives the smallest (acute) angle that satisfies the condition within this quadrant.
By understanding that the result of \( \arccos\left(\frac{1}{2}\right) \) yields an angle in the first quadrant, you can confidently find the measure of \( \theta \) as an acute angle.