Problem 16 - Precalculus : Building Concepts and Connections

Question

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$B=75^{\circ}, C=50^{\circ}, b=25$$

Step-by-Step Solution

Answer

The solved triangle has the following measures: A = 55 degrees, a = 21.6025 units, c = 19.6194 units.

Step 1: Calculate the Remaining Angle

First, the remaining angle A can be calculated using the principle that the sum of angles in a triangle equals \(180^\circ\). Thus, the formula to find the unknown angle is \(A = 180^\circ - B - C = 180^\circ - 75^\circ - 50^\circ = 55^\circ\).

Step 2: Use the Law of Sines to Find a

Next, the Law of Sines can be used to find side 'a'. The formula \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\) simplifies to \(a = \sin(A) * \frac{b}{\sin(B)}\), substututing the given values gives: \(a = \sin(55^\circ) * \frac{25}{\sin(75^\circ)}\). Using a scientific calculator to compute the sine calculations and round the answer to four decimal places, we find that 'a' approximately equals 21.6025.

Step 3: Use the Law of Sines to Find c

Using the same Law of Sines principle, we can find 'c'. The formula \(\frac{c}{\sin(C)} = \frac{b}{\sin(B)}\) simplifies to \(c = \sin(C) * \frac{b}{\sin(B)}\), substututing the given values gives: \(c = \sin(50^\circ) * \frac{25}{\sin(75^\circ)}\). Implementing the sine calculations and rounding the result to four decimal places, 'c' approximately equals 19.6194.

Key Concepts

Law of Sines

When tasked with solving a triangle, the Law of Sines is an essential tool for relating the angles and sides of non-right triangles. This law states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle. Specifically, the formula is expressed as:
\[\begin{equation}\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\end{equation}\]where 'a', 'b', and 'c' represent the triangle's sides, and 'A', 'B', and 'C' are the respective opposite angles.
In practice, this law allows us to solve for missing sides or angles given at least one side-length and its opposite angle, along with another angle or side. For example, in our exercise, once we determine the third angle, 'A', we can use the Law of Sines to find the unknown sides 'a' and 'c' by manipulating the formula to isolate the missing side.

Triangle Angle Sum

Another fundamental concept in triangle geometry is the triangle angle sum theorem. This theorem posits that the sum of the angles within a triangle will always add up to 180 degrees. This is expressed in the formula:
\[\begin{equation}A + B + C = 180^\circ\end{equation}\]where 'A', 'B', and 'C' represent the triangle's interior angles.
Knowing this, we can easily deduce the size of an unknown angle when the other two are known, which is what we do in step 1 of our provided solution. Determining the remaining angle, 'A', is crucial because it is required when applying the Law of Sines. To ensure complete understanding of the relationships inside triangles, grasping this theorem is critical.

Scientific Calculator

Working with triangles often involves calculations with trigonometric functions, which can be complex and are best handled with a scientific calculator. These calculators are designed to compute functions, like sine, cosine, and tangent, which are frequently needed for solving triangles, especially when using the Law of Sines.
To correctly use a scientific calculator for our purposes, ensure it is set to the correct mode (degree mode, in this case), then simply input your expressions as needed. For our triangle problem, after finding angle 'A', we calculate 'a' and 'c' by taking the sines of the angles and dividing them by the sine of another known angle, all done with the aid of a scientific calculator. When performing these calculations, it's vital to round the results to the desired decimal places as stated in the problem or by convention for your field of study.