Problem 45 - Algebra 2 and Trigonometry

Question

In \(39-50,\) find the smallest positive value of \(\theta\) to the nearest degree. $$ \cos \theta=0.9990 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is \( 3 \) degrees.

Step 1: Understand the Problem

We are given the equation \( \cos \theta = 0.9990 \) and need to find the smallest positive value of \( \theta \) that satisfies this equation. The answer should be accurate to the nearest degree.

Step 2: Recall the Range of Cosine Function

The function \( \cos \theta \) has a range of values from -1 to 1. The cosine function is periodic with a period of 360 degrees, meaning the pattern repeats every 360 degrees.

Step 3: Using Inverse Cosine Function

To find \( \theta \), we can use the inverse cosine function, denoted as \( \arccos \) or \( \cos^{-1} \). So, \( \theta = \arccos(0.9990) \).

Step 4: Calculate \( \theta \) Using a Calculator

Using a scientific calculator, compute the inverse cosine of 0.9990: \( \theta = \arccos(0.9990) \approx 2.57 \) degrees.

Step 5: Round to the Nearest Degree

Round the calculated angle \( \theta \approx 2.57 \) to the nearest whole number. This gives us \( \theta = 3 \) degrees.

Key Concepts

Cosine Function

The cosine function is one of the basic trigonometric functions crucial in mathematics, particularly in analyzing angles and triangles. It relates to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Essentially, the cosine of an angle in a right triangle is the length of the side adjacent to the angle divided by the length of the hypotenuse.
This function is represented as \( \cos \theta \), where \( \theta \) is the angle. The cosine function is periodic, which means its values repeat in a regular cycle. The period of the cosine function is 360 degrees or \( 2\pi \) radians, suggesting that every 360 degrees, the function's values replay themselves.

• Range: The values of the cosine function range from -1 to 1.
• Graph: The cosine graph is a wave that begins at 1 when \( \theta = 0 \) and oscillates between -1 and 1.
• Applications: It's widely used in physics, engineering, and various applications including calculating distances and angles.

Understanding the cosine function is imperative for manipulating trigonometric expressions and solving related equations, like finding angles when a specific cosine value is known.

Inverse Trigonometric Functions

Inverse trigonometric functions are operations that allow us to find an angle given a known trigonometric ratio. The inverse cosine function, indicated as \( \arccos \) or \( \cos^{-1} \), lets us find an angle when we know the cosine value.
In trigonometry, when we use \( \arccos(x) \), we look for the angle \( \theta \) such that \( \cos \theta = x \). This inverse function returns an angle, primarily in the range of 0 to 180 degrees for real numbers between -1 and 1, which is important in many practical applications.

• Domain: The input value for \( \arccos \) must be within the range of -1 to 1.
• Output: The result is an angle, often represented in degrees or radians.
• Usage: Inverse trigonometric functions are used to solve equations where an angle needs to be determined from a trigonometric ratio.

For instance, in our exercise, we used the inverse cosine function to find \( \theta \) such that \( \cos \theta = 0.9990 \).

Degrees

In trigonometry, degrees are a measurement unit for angles. Understanding angles in degrees is fundamental when discussing angle measures in various contexts, particularly in mathematics and geometry. One full circle is divided into 360 degrees, making it a convenient and widely used system for angle measurement.
Conversions between degrees and other units like radians are common. One complete revolution (360 degrees) equates to \( 2\pi \) radians.

• Full circle: 360 degrees.
• Straight angle: 180 degrees.
• Right angle: 90 degrees.

Degrees allow for straightforward computations in geometry and trigonometry. When solving the problem, we rounded the angle value to the nearest degree after calculating \( \theta = \arccos(0.9990) \). Understanding degrees helps in interpreting results in real-world applications or when visualizing angles.