Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arctan \(\frac{7}{2}\)

Step-by-Step Solution

Answer

The approximate value of arctan(7/2), rounded to the nearest hundredth, is 1.29 radians.

Step 1: Identify the expression

The expression given is the inverse tangent of 7 over 2, written as arctan(7/2)

Step 2: Use a scientific calculator

Enter the expression, arctan(7/2), into your scientific calculator. Be sure your calculator is set to output the result in radians if it has the option to choose between degrees and radians.

Step 3: Round to the nearest hundredth

Once you get the result from the calculator, round it to the nearest hundredth to get your final answer.

Key Concepts

Arctan Function

The arctan function, often written as \( \text{arctan}(x) \) or \( \tan^{-1}(x) \), is an inverse trigonometric function. It is used to determine the angle whose tangent is a given number. In trigonometry, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side.
For example, if the tangent of angle \( \theta \) is \( \frac{7}{2} \), then \( \theta = \text{arctan}(\frac{7}{2}) \). This function is essential because it helps us find the measure of an angle when we know its tangent ratio.
One important thing to understand about arctan is its range. The result of an arctan function is usually given in radians, and it is constrained from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), which corresponds to angles from -90° to 90°.

Scientific Calculator Use

To solve problems involving the arctan function, like \( \text{arctan}(\frac{7}{2}) \), a scientific calculator is invaluable. Most calculators have a designated button for arctan, often marked as 'tan⁻¹'.
Here's a simple guide on how to use your scientific calculator:

• First, make sure your device is set to the correct angle measurement. Check whether the problem requires your answer in radians or degrees.
• Input the expression you wish to calculate. For arctan, this usually means pressing the 'tan⁻¹' button and then entering the fraction or decimal value.
• Press 'Enter' or the 'Equals' button to get your result.

Remember that calculators vary slightly, so it's a good idea to familiarize yourself with your specific model. Checking the user manual or online resources might provide helpful tips.

Rounding Numbers

Rounding numbers is a critical skill whenever you deal with calculations or measurements. For the arctan function, once the calculator provides the output, you often need to round it to a specified degree of precision. In our example, we're asked to round to the nearest hundredth.
To round a number to the nearest hundredth (two decimal places), identify the second decimal place. Look at the third decimal place, which is right next to it. If this number is 5 or greater, you round up the second decimal place. If it is less than 5, you leave the second decimal place as it is.
For example, if your calculator gives a result of 1.35729 for \( \text{arctan}(\frac{7}{2}) \), you would round it to 1.36 since the third decimal is a 7.

Radians and Degrees

Radians and degrees are two units of measuring angles. Understanding the difference between them is vital, especially when dealing with trigonometric functions.
- **Degrees** are a more intuitive way for many people, usually coming from general education. A full circle is 360 degrees, meaning very familiar everyday units like 90° or 180°.
- **Radians** are often used in professional mathematics and sciences. They are based on the radius of a circle. One full circle (360°) is equivalent to \( 2\pi \) radians.
When solving arctan problems, your calculator might be set to either of these units. Always ensure that your calculator is in the mode you wish to use.
Converting between these two is simple: Use\( \pi \, \text{radians} = 180° \), so to convert degrees to radians, multiply by \( \frac{\pi}{180} \), and to convert radians to degrees, multiply by \( \frac{180}{\pi} \).
For our example of \( \text{arctan}(\frac{7}{2}) \), knowing how to switch or convert between these units could help understand and verify your results.