Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(A=68.41^{\circ}, B=54.23^{\circ}, a=12.75 \mathrm{feet}\)

Step-by-Step Solution

Answer

Find \(C = 57.36^{\circ}\); use Law of Sines to get \(b = 11.06\) ft, \(c = 11.30\) ft.

Step 1: Find Missing Angle C

To find angle \(C\), use the triangle angle sum property which states that the sum of the angles in a triangle is \(180^{\circ}\). Thus, \(C = 180^{\circ} - A - B = 180^{\circ} - 68.41^{\circ} - 54.23^{\circ}\). Calculate the remaining angle \(C\).

Step 2: Apply the Law of Sines

Use the Law of Sines which is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Since \(a = 12.75\), \(A = 68.41^{\circ}\) and \(B = 54.23^{\circ}\), and you already found \(C\), use \( \frac{a}{\sin A} = \frac{b}{\sin B} \) to find \(b\).

Step 3: Calculate Side b

Rearrange the Law of Sines to solve for \(b\): \( b = \frac{a \cdot \sin B}{\sin A} \). Substitute the known values to calculate \(b\).

Step 4: Calculate Side c

Now, use the equation \( \frac{a}{\sin A} = \frac{c}{\sin C} \) from the Law of Sines to find \(c\): \( c = \frac{a \cdot \sin C}{\sin A} \). Substitute the known values to calculate \(c\).

Key Concepts

Law of Sines

The Law of Sines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. This law is particularly useful when dealing with non-right triangles. The formula is expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
This equation allows us to find missing side lengths or angles in a triangle when given enough information. For the triangle given in the exercise, since you have angle \(A\), angle \(B\), and side \(a\), you can rearrange the formulas to solve for sides \(b\) and \(c\). Just plug in the known values, ensuring that you use the sine function accurately from a scientific calculator. This powerful tool helps us solve various triangle-related problems effectively.

Triangle Angle Sum Property

The Triangle Angle Sum Property is a simple yet very important rule. It states that the sum of the angles in any triangle is always \(180^{\circ}\). This property is essential when solving triangles because it helps you determine the third angle when you know the other two.

• For example, if you know angles \(A\) and \(B\), you can find angle \(C\) by subtracting the sum of \(A\) and \(B\) from \(180^{\circ}\).
• In the exercise, having \(A = 68.41^{\circ}\) and \(B = 54.23^{\circ}\), you can calculate \(C\) as follows: \(C = 180^{\circ} - 68.41^{\circ} - 54.23^{\circ}\).

This practical property ensures you always have a method to find missing angles, paving the way to further use of tools like the Law of Sines for finding unknown side lengths.

Trigonometry

Trigonometry, the branch of mathematics dealing with the relationships between the angles and sides of triangles, plays a crucial role here. It involves using various trigonometric functions, like sine, cosine, and tangent, to solve for unknowns in a triangle.
In the provided problem:

• The sine function is primarily used, which relates angles to the ratio of opposite sides in a triangle.
• These calculations with sine let us decide the lengths of sides when angles are known and vice versa.

Understanding these core trigonometric principles increases accuracy and confidence in solving many geometrical problems, enhancing problem-solving skills.
Trigonometry aids in accurate measurements and has broad applications in fields such as physics, engineering, and computer graphics.

Angles in a Triangle

Every triangle is defined by its three angles. Knowing how these angles interact and sum up is crucial for solving triangles. In simpler terms:

• The sum of the interior angles is always \(180^{\circ}\).
• Each angle plays a unique role; knowledge of two angles ensures you can easily find the third.

The exercise illustrates that if you have two angles, you immediately grasp the third by simple subtraction. This concept is fundamental not only for solving triangles but also for understanding more complex geometrical shapes. This basic understanding lays the groundwork for more advanced topics in geometry and trigonometry, reminding us of how interconnected these mathematical theories are.