Problem 40 - Basic Technical Mathematics with Calculus

Question

Find the indicated part of the right triangle that has the given parts. The legs are 0.596 and \(0.842 .\) Find the larger acute angle.

Step-by-Step Solution

Answer

The larger acute angle is approximately 54.18 degrees.

Step 1: Understand the Problem

We have a right triangle with legs of lengths 0.596 and 0.842. We need to find the larger acute angle, referring to the angle opposite the longer leg of the triangle.

Step 2: Identify Relevant Trigonometric Formula

To find an angle in a right triangle given two sides, we can use the tangent function: \[\tan(\theta) = \frac{\text{opposite leg}}{\text{adjacent leg}}\]For the larger acute angle, the opposite side is the longer leg (0.842) and the adjacent side is the shorter leg (0.596).

Step 3: Calculate the Tangent of the Angle

Calculate the tangent of the larger acute angle using the formula: \[\tan(\theta) = \frac{0.842}{0.596}\]Calculate this fraction to get the value for \(\tan(\theta)\).

Step 4: Use Inverse Trigonometric Function

To find the angle \(\theta\), take the inverse tangent (arctan) of the calculated tangent value. This is denoted as: \[\theta = \tan^{-1}\left(\frac{0.842}{0.596}\right)\]Use a calculator to find \(\theta\) in degrees.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that includes a 90-degree angle. This is known as the right angle. The sides that form this right angle are called the legs, and the side opposite the right angle is the longest side, known as the hypotenuse. In this exercise, we are dealing with the legs of the right triangle which are given as 0.596 and 0.842.
The hypotenuse is not mentioned, because we are focusing on finding an angle rather than the sides.

• The side opposite the larger acute angle is always the longer leg in a right triangle.
• The acute angles in a right triangle are those angles that are less than 90 degrees.

Understanding how these parts relate to each other helps us apply the correct formulas when solving problems.

Tangent Function

The tangent function plays a crucial role in right triangles, allowing us to find angles when we know the length of the opposite and adjacent sides. This function is defined as the ratio of the length of the opposite side to the length of the adjacent side.For a given angle \(\theta\), the tangent is expressed as:

\[ \tan(\theta) = \frac{\text{opposite leg}}{\text{adjacent leg}} \]

In our case, to find the larger acute angle, we use the longer leg as the opposite side (0.842) and the shorter leg as the adjacent side (0.596). Calculating this ratio helps us move one step closer to finding \(\theta\).

• The tangent provides a simple way to relate the angle to the sides of the triangle.
• Remember, the larger the tangent value, the larger the angle, as both the tangent function and the angle in radians increase.

Inverse Trigonometric Functions

After calculating the tangent of an angle, the next step is to find the actual angle measure from this tangent value. This is where the inverse trigonometric functions come in, specifically the inverse tangent, which is often written as \(\tan^{-1}\) or "arctangent."When you have \( \tan(\theta) = x \), you find \( \theta \) by computing \( \tan^{-1}(x) \).
In our scenario, we found the tangent as \( \frac{0.842}{0.596} \). By inputting this value into your calculator as \( \tan^{-1} \), you obtain the angle \( \theta \) in degrees.

• The inverse tangent function is essential as it translates the ratio back into a degree measure that can be easily understood.
• This function is useful for any right triangle problem involving angle measures.

Once you find the larger acute angle, you've effectively solved the problem using trigonometric functions.