Problem 15 - Algebra 2

Question

Find each angle measure to the nearest tenth of a degree. \(\tan ^{-1} 3.552\)

Step-by-Step Solution

Answer

The angle measure to the nearest tenth of a degree is approximately 74.5 degrees.

Step 1: Understand Inverse Tangent

The inverse tangent function, also known as arctangent or \(\tan^{-1}\), is used to find the angle from the tangent of that angle. It's the 'opposite' of the tangent function.

Step 2: Apply Inverse Tangent

Apply the inverse tangent function to the given value, 3.552. This is done by feeding 3.552 into your calculator's \(\tan^{-1}\) function (usually accessed by pressing 2nd or Shift followed by the tangent button). Make sure your calculator is in degree mode.

Step 3: Round the Result

Since we are asked to find the angle measure to the nearest tenth of a degree, round off the result from step 2 to one decimal place.

Key Concepts

Understanding Arctangent

The arctangent is an essential concept in trigonometry, often denoted as \( \tan^{-1} \). It allows us to determine the angle whose tangent is a given number. Imagine the tangent function as a mountain slope. The tangent value represents the steepness of the slope, while the arctangent helps us find the angle of elevation. This is why we refer to arctangent as the 'inverse' of the tangent function. Instead of providing the ratio of the opposite to the adjacent side, arctangent takes that ratio and gives us an angle. Arctangent is particularly useful in fields such as physics and engineering where calculating angles based on given tangential values is necessary. By understanding and using the arctangent function correctly, you can accurately determine angle measures for various applications.

Angle Measurement in Degrees

When measuring angles, one of the most common units used is degrees. A full circle is divided into 360 degrees, much like a clock face but in a complete loop. Degrees provide a practical way to denote angles that are simple to understand, especially for basic trigonometry problems.Remember, when using trigonometric functions on a calculator, it's crucial to ensure your calculator is set to 'degree mode' as opposed to 'radians,' which is another measurement unit where a full circle equals \(2\pi\) radians. Selecting the incorrect mode can lead to answers that are significantly off. It’s like using the wrong currency—everything looks right, but the value doesn't make sense.

Rounding to the Nearest Tenth

Rounding numbers is a necessary skill, especially when you're dealing with decimal places in trigonometry. We often round to simplify calculations and to fit results into a more usable form. When tasked with rounding a number to the nearest tenth, observe the digit in the hundredth place (second place after the decimal). Here’s a simple step-by-step approach:

• If the hundredth digit is 5 or greater, round the tenth digit up by one.
• If it's less than 5, leave the tenth digit as it is.

This process helps to communicate results more clearly and concisely, making your answers easier to interpret. In scientific and everyday calculations, rounding helps maintain precision without unnecessary complexity. This is especially helpful in cases like finding angle measures where values are constantly applied in further calculations.