Problem 48 - Algebra 2 and Trigonometry

Question

In \(39-50,\) find the smallest positive value of \(\theta\) to the nearest degree. $$ \cos \theta=0.5446 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is 57 degrees.

Step 1: Understand the Problem

We need to find the smallest positive angle \( \theta \) such that the cosine of the angle is equal to 0.5446, and then round this angle to the nearest degree.

Step 2: Use the Cosine Inverse Function

The cosine inverse function, \( \cos^{-1} \), is used to find an angle whose cosine is a given number. Calculate \( \theta = \cos^{-1}(0.5446) \).

Step 3: Calculate Using a Calculator

Input the value \( 0.5446 \) into a scientific calculator and use the inverse cosine function to find the angle \( \theta \). This gives \( \theta \approx 57.098 \) degrees.

Step 4: Round to the Nearest Degree

Since \( \theta \approx 57.098 \) degrees, we round this to the nearest degree, which results in \( \theta = 57 \) degrees.

Key Concepts

Cosine Inverse Function

The cosine inverse function, denoted as \( \cos^{-1} \), helps to determine the angle \( \theta \) if you know its cosine value. This is often necessary in trigonometry when solving for angles in various kinds of problems. The function is also known as "arc cosine," and it is the inverse operation of the cosine function. The range of \( \cos^{-1} \) is from 0 to 180 degrees, making it specifically useful for finding the smallest positive angle in degrees.
When given a specific cosine value, say 0.5446, the goal is to use \( \cos^{-1} \) to compute the precise angle measure that, when its cosine is calculated, yields back 0.5446. This will typically be done using a calculator, as solving such equations by hand can be impractical.

Rounding to Nearest Degree

Once you've determined the angle using the cosine inverse function, you often need to round it to make it more usable or meaningful, especially when dealing with degrees. In this context, rounding to the nearest degree involves:

• Identifying the decimal part of the angle.
• If this decimal part is 0.5 or higher, you round up.
• If the decimal part is less than 0.5, you round down.

For instance, after calculating \( \theta \approx 57.098 \) degrees, the rounding process would focus on 0.098. Since this is less than 0.5, \( \theta \) is rounded down to 57 degrees. This step ensures precision in reporting your results, making it easier to communicate within the context of practical applications.

Scientific Calculator Usage

Scientific calculators are essential tools for solving trigonometric problems, especially those involving inverse functions. Here's a simple guide for using one:To find an angle using the arc cosine, follow these steps:

• Turn on your calculator and switch to degree mode (as opposed to radians).
• Identify the function keys on your calculator. Look for the key marked \( \cos^{-1} \).
• Input the cosine value you have, for example, 0.5446.
• Press the \( \cos^{-1} \) key to compute the angle \( \theta \).
• Read the resulting angle, which, for our exercise, is approximately 57.098 degrees.

Make sure the calculator is properly set to degrees. Otherwise, the output will not reflect the intended context, which could lead to errors in further calculations or interpretations.