Problem 37 - Precalculus Essentials

Question

$$\text {use a calculator to find the value of the acute}\text { angle } \theta \text { to the nearest degree.}$$ $$\tan \theta=4.6252$$

Step-by-Step Solution

Answer

The value of the acute angle \( \theta \) to the nearest degree is obtained by finding \( \theta = \arctan(4.6252) \) using a calculator, ensuring the answer is in degrees mode, and rounding off to the nearest degree.

Step 1: Understanding the Problem

We're given \( \tan \theta = 4.6252 \) and are asked to find the value of \( \theta \) (in degrees) which is an acute angle. Since the value of tangent is positive and greater than one, we know that \( \theta \) should lie between 45° and 90°.

Step 2: Using the Calculator

We can directly find the acute angle in degrees using the formula \( \theta = \arctan(4.6252) \). On a calculator, input the tangent value, which is 4.6252, then apply the inverse tangent or arctan function. Make sure the calculator mode is set to degrees, not radians, to get the answer in degrees.

Step 3: Rounding Off

Once the calculator gives the result, the answer is rounded to the nearest degree as per the requirements of the problem.

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry, relating to angles and their positions in right-angled triangles. In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. This means that, given an angle \( \theta \),

• \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \)

This relationship is essential in figuring out the angle itself or any of the sides of the triangle. The tangent function is periodic, repeating every \(180^\circ\) or \( \pi \text{ radians} \), and it approaches infinity at \(90^\circ\) or \(\frac{\pi}{2} \text{ radians} \).
If the value of \( \tan(\theta) \) is known, as in the given problem where \( \tan(\theta) = 4.6252 \), we use this information to find the angle \( \theta \). Since the tangent value is positive, the angle \( \theta \) must be in the first quadrant where the tangent is positive and less than \(90^\circ\).
Understanding the tangent function helps to solve problems related to angles and trigonometric identities more easily.

arctan (inverse tangent)

The arctan function, also written as the inverse tangent or \( \tan^{-1} \), is used to find an angle when the tangent value is given. It essentially reverses the tangent function.
When you use \( \arctan \), you're looking for the angle whose tangent value is known. For example, in the exercise given:

• We are given \( \tan(\theta) = 4.6252 \)
• To find \( \theta \), we compute \( \theta = \arctan(4.6252) \)

Typically, calculators can compute arctan directly, and it's crucial to ensure the calculator is in degree mode to match the required format of the answer.
The inverse tangent function only returns angles between \(-90^\circ\) and \(90^\circ\) or \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, ensuring that what you calculate is an acute angle or another angle within this range. In this specific problem, since we know the tangent value is quite high, \( \theta \) will be closer to \(90^\circ\).
This function is a powerful tool, especially when paired with a calculator, for determining angle measures in real-world applications.

Acute Angle

An acute angle is any angle that measures less than \( 90^\circ \). These angles are common in trigonometry and in solving problems involving right-angled triangles. In our current problem, we are tasked with finding such an angle \( \theta \) given a tangent value.
Acute angles are important because they provide insight into the relative proportions of sides in a triangle. For instance, if you know that the tangent is the ratio of the opposite side to the adjacent side, then having a large tangent value like 4.6252 indicates a larger opposite side compared to the adjacent, suggesting an angle closer to the right angle itself (\( 90^\circ \)).
With this specific problem, since the requirement is to find an angle accurate to the nearest degree, and because \( \theta \) is supposed to be an acute angle, it confirms the need for using appropriate functions, like \( \arctan \), to calculate and validate that the desired angle is indeed acute. This ensures clarity and correctness in the solution process.