Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-23,\) use a calculator to find each value of \(\theta\) to the nearest degree. $$ \theta=\arctan 4.4 $$

Step-by-Step Solution

Answer

\( \theta \approx 77^\circ \)

Step 1: Understand the ArcTan Function

The arctan function, denoted as \( \arctan \), is the inverse of the tangent function. It is used to find the angle for a given tangent ratio. Therefore, \( \theta = \arctan(4.4) \) means we need to find the angle whose tangent is 4.4.

Step 2: Use a Calculator to Find θ

Using a scientific calculator, enter the arctan or tan⁻¹ function to calculate the angle \( \theta \). Input the value 4.4 as: \[ \theta = \arctan(4.4) \approx 77.29181... \]

Step 3: Round θ to the Nearest Degree

To find the angle \( \theta \) to the nearest degree, round the calculated value 77.29181 to the nearest whole number. Since the decimal is 0.29181, we round down to 77 degrees.

Key Concepts

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles from known ratios of sides in right-angled triangles. When you hear about functions like the arctangent (denoted as \(\arctan\)), it's all about reversing the tangent function. While the regular tangent function takes an angle and gives you the ratio of opposite side to adjacent side, the arctangent function does the reverse. So when you see \( \arctan(4.4) \), what you’re getting is the angle \( \theta \) whose tangent value is 4.4. Inverse trigonometric functions like \( \arctan \) are handy when you're dealing with trigonometry problems that provide you with side ratios and ask you to figure out the angle. They act as the essential bridge between side lengths and angles in triangles.

Rounding Numbers

Rounding numbers is a simple yet powerful tool in mathematics. It allows us to simplify numbers and make them more manageable either for communication or for performing further calculations. The concept is about altering a number to a nearby, more convenient value. Here's how it generally works:

• If the decimal part of a number is less than 0.5, you round down.
• If the decimal part is 0.5 or more, you round up.

For example, when you're given \(77.29181\) and asked to round it to the nearest degree, you check the decimal part, which is \(0.29181\). Since it is less than \(0.5\), you round down. This means \(\theta\) becomes \(77\) degrees without that complication of decimals. Rounding helps us when we need cleaner numbers, especially when precision is not crucial, like expressing angles in degrees.

Using a Calculator in Trigonometry

Calculators are pivotal when working with trigonometric functions and angles, particularly in more complex calculations. Here's a brief guide on using a calculator to find inverse trigonometric functions:- **Switch to Deg/Rad Mode**: Make sure your calculator is set to degree mode if you're working with degrees, or radian mode if radians are needed.
- **Locate the Function**: Find the \(\arctan\) or \(\tan^{-1}\) function. Often, it's a secondary feature accessible through the "shift" or "2nd" button on your calculator.
- **Input the Value**: Enter the ratio value, in this case, \(4.4\).
- **Calculate**: Hit "enter" or "=" to compute the angle \(\theta\).Using a calculator correctly ensures you get quick, precise answers, which is essential when prepping for exams or solving homework. It minimizes errors and speeds up your solving process, letting you focus on understanding concepts rather than arithmetic.