Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-23,\) use a calculator to find each value of \(\theta\) to the nearest degree. $$ \theta=\arccos 0.9 $$

Step-by-Step Solution

Answer

\( \theta = 26 \) degrees.

Step 1: Understand the Problem

We need to find the angle \( \theta \) where the cosine of \( \theta \) is \( 0.9 \). The angle \( \theta \) will be the result of the inverse cosine function, written as \( \theta = \arccos(0.9) \).

Step 2: Use a Calculator to Find \( \arccos(0.9) \)

Input the value \( 0.9 \) into the calculator to find the angle \( \theta \) using the inverse cosine function. Make sure that the calculator is set to degrees rather than radians.

Step 3: Record the Calculator Result

Once calculated, the inverse cosine of \( 0.9 \) gives \( \theta \approx 25.84 \) degrees. We will round this value to the nearest degree.

Step 4: Round \( \theta \) to the Nearest Degree

Round the result from Step 3, \( 25.84 \), to the nearest whole number. As the decimal part is 0.84, we round up, making it \( 26 \) degrees.

Key Concepts

Understanding the Arccosine Function

The arccosine function, denoted as \( \arccos \), is a type of inverse trigonometric function that uncovers the angle whose cosine is a given number. In simpler terms, if you have a known cosine value and you want to find the angle that produces this value when cosine is applied, you use \( \arccos \). This is crucial when working in real-world scenarios and many areas of mathematics and physics, such as solving triangles. The range of the \( \arccos \) function in degrees is 0° to 180°, meaning any output angle will be within this interval.

• It flips the role of angles and cosine values: from inputting an angle to find its cosine, to inputting a cosine value to find the angle.
• Key for calculation purposes, especially in obtaining specific angles from given ratios.

Understanding Degree Measure

Angles can be measured in degrees or radians, but degrees are commonly used in everyday applications. A full circle is 360°, and this measure helps us easily visualize and understand the size of an angle. Knowing whether your calculator works in degrees or radians is crucial for calculations, as using the wrong setting can lead to incorrect results.

• The degree measure is based on dividing a circle into 360 equal parts.
• It is very intuitive and aligns well with angles we commonly encounter in different areas, from geometry to navigation.

Effective Use of Calculators

When dealing with trigonometric functions like \( \arccos \), calculators are incredibly handy for performing precise calculations. To find \( \arccos(0.9) \), ensure your calculator is set to degree mode—otherwise, you might end up with a result in radians. Many scientific calculators will have a dedicated button for \( \arccos \) or provide it in a menu.

• Ensure your calculator is in the correct mode (degrees in this case).
• Enter the numerical value you know (here, 0.9) and then apply the \( \arccos \) function.
• Confirm the output makes sense within the expected range of 0° to 180°.

The Importance of Rounding Numbers

Rounding numbers is the process of simplifying a number while keeping its value close to what it was. This is particularly useful in scenarios where precision to many decimal places isn't necessary, and a whole number is more convenient, like when reporting angles. For example, if you find \( \theta \approx 25.84 \) degrees, the conventional way to round is to check the decimal place:

• If the decimal is 0.5 or higher, round up.
• If it's less than 0.5, round down.

Thus, \( 25.84 \) becomes \( 26 \) degrees when rounded to the nearest degree, as 0.84 instructs us to round up.