Problem 38 - Basic Technical Mathematics with Calculus

Question

Find the indicated part of the right triangle that has the given parts. One leg is \(8.50,\) and the angle opposite this leg is \(52.3^{\circ} .\) Find the other leg.

Step-by-Step Solution

Answer

The other leg is approximately 6.64 units long.

Step 1: Identify the Given Parts

We have a right triangle with one leg measuring 8.50 units and an angle opposite to this leg measuring \( 52.3^{\circ} \). Our task is to find the length of the other leg.

Step 2: Determine the Relationship

In a right triangle, the relationship between an angle and its opposite side and adjacent side is determined by the tangent function, given by \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 52.3^{\circ} \) and the opposite side (one leg) is 8.50.

Step 3: Set Up the Tangent Equation

Using the tangent function: \[ \tan(52.3^{\circ}) = \frac{8.50}{x} \]where \( x \) is the length of the other leg.

Step 4: Solve for the Other Leg

Rearrange the equation to solve for \( x \):\[ x = \frac{8.50}{\tan(52.3^{\circ})} \]Calculate the value:\[ x \approx \frac{8.50}{1.2799} \approx 6.64 \]Hence, the length of the other leg is approximately 6.64 units.

Key Concepts

Understanding the Right Triangle

A right triangle is a type of triangle that has one of its angles measuring exactly 90 degrees. This particular angle is known as the right angle. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are known as the legs. They can be identified as the adjacent and opposite sides, relative to the angle of interest.
Recognizing these parts is crucial because different trigonometric functions relate these sides based on the angles within the triangle. When given a problem involving a right triangle, it's important to first identify the right angle, the hypotenuse, and determine which of the remaining sides are the opposite and adjacent sides relative to the angle you are working with.

Explaining the Tangent Function

The tangent function is one of the primary trigonometric functions used in studying triangle relationships. It is usually abbreviated as 'tan' and relates the angles to the ratio of the lengths of the sides. Particularly, it concerns the opposite and adjacent sides of a right triangle.
The tangent of an angle is defined as:

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

Where:

• Opposite: The side opposite the angle \(\theta\).
• Adjacent: The side next to the angle \(\theta\), but not the hypotenuse.

When you know the length of one leg and an angle (other than the right angle), you can use the tangent function to find the other leg. This equation is valuable because it allows you to solve for the unknown side as long as one side and one angle are specified.

Trigonometric Calculations in Practice

Applying trigonometric calculations in problems involves a step-by-step approach to using the appropriate trigonometric function. First, identify what is known and what needs to be found. Choosing the correct function, whether it be sine, cosine, or tangent, is key to finding a solution.
In our example, given an angle and the opposite side, the tangent function was the best fit. By setting up the trigonometric equation (\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)), you can rearrange it to solve for the unknown side.
By plugging in the known values:

• The known leg (opposite) is 8.5 units long.
• The angle is \( 52.3^\circ \).

The calculation steps were:

• Set up the equation: \( \tan(52.3^\circ) = \frac{8.5}{x} \).
• Rearrange to solve for \( x \): \( x = \frac{8.5}{\tan(52.3^\circ)} \).
• Calculate \( \tan(52.3^\circ) \approx 1.2799 \) using a calculator.
• Solve \( x \approx \frac{8.5}{1.2799} \approx 6.64 \).

This step-by-step process allows you to solve for unknown triangle dimensions using basic trigonometry.