Problem 61 - Algebra and Trigonometry

Question

Use a calculator to find the value of the acute angle \(\theta\) in radians, rounded to three decimal places. $$ \tan \theta=0.4169 $$

Step-by-Step Solution

Answer

The acute angle \(\theta\) is approximately \(0.397\) radians when rounded to three decimal places.

Step 1: Understand the Problem

We are given that the tangent of an acute angle \(\theta\) is 0.4169. This means that for a right triangle with angle \(\theta\), the ratio of the length of the side opposite to \(\theta\) to the length of the side next to \(\theta\) (that is, the ratio of the height to the base when considering a right triangle) is 0.4169. Our job is to find the corresponding acute angle \(\theta\) in radians.

Step 2: Use the Inverse Tangent Function

We can use the inverse tangent function, denoted as `arctan` or `tan^{-1}`, to find the angle \(\theta\) given its tangent value. Applying the `arctan` function to both sides of the expression \(\tan \theta = 0.4169\), we get \(\theta = \arctan(0.4169)\).

Step 3: Use a Calculator to Compute the Answer.

Most calculators should have the arctan (or tan^-1) function. Set your calculator to radian mode, then calculate \(\theta = \arctan(0.4169)\).

Key Concepts

Tangent of an Angle

When we talk about the tangent of an angle in the context of trigonometry, we are describing a fundamental relationship within right triangles. The tangent of an angle, often abbreviated as 'tan', is a ratio. It's the measure of the side opposite the angle of interest, divided by the side adjacent to the angle within a right-angled triangle. In mathematical terms, for an angle \( \theta \), the tangent is represented as \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \).

Understanding the tangent is essential since it is one of the basic trigonometric functions that helps us relate the angles of a triangle to its side lengths. This concept is particularly useful in various fields such as physics, engineering, astronomy, and even in the simple task of finding the height of inaccessible objects using indirect measurement.

Acute Angle in Radians

Radians are the standard unit of angular measure used in many areas of mathematics. An acute angle is one that is less than 90 degrees (or \( \frac{\pi}{2} \) radians). Unlike degrees, which divide a circle into 360 arbitrary units, radians are based on the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle.

Typically, when we work with trigonometric functions, we can use either degrees or radians, but radians offer a more natural approach to calculating angles in terms of the properties of the circle. A full circle corresponds to \( 2\pi \) radians, which means that converting an acute angle from degrees to radians requires multiplying by \( \frac{\pi}{180} \). Conversely, to convert from radians back to degrees we multiply by \( \frac{180}{\pi} \).

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, which, fundamentally, is the study of the relationships between angles and sides of triangles. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each referencing the ratio of sides in a right triangle relative to one of its angles.

Other functions include secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)), which are the reciprocals of cosine, sine, and tangent, respectively. These functions are invaluable for solving problems involving triangles, modeling periodic phenomena, and in managing waves, oscillations, and many other applications across scientific domains.

Using a Calculator for Trigonometry

Modern calculators are equipped with functions to aid in solving trigonometric calculations, which are fundamental when you're dealing with angles and distances. Using a calculator for trigonometry typically involves switching the mode from degrees to radians if necessary, according to the context of the problem.

For tangents and their associated calculations, calculators use the inverse tangent function, often denoted as \( \arctan \) or \( \tan^{-1} \), which allows you to find the angle when you know the tangent ratio. It's important for students to become familiar with these functions, ensuring they are set correctly for the task at hand, and to interpret the results within the correct units (degrees or radians). By mastering the use of these calculator functions, solving trigonometry problems can become much quicker and more accurate.