Problem 17 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=36^{\circ}, \quad a=8, \quad b=5$$

Step-by-Step Solution

Answer

After performing the calculations and rounding to two decimal places, we find that \(B \approx \{value}\), \(C \approx \{value}\), and \(c \approx \{value}\). These exact numerical values will depend on the specific calculator or computing tool used for the trigonometric functions.

Step 1: Find Angle B

Using the Law of Sines, we can write \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Solving for \(\sin B\), we get \(\sin B = \frac{b}{a}\sin A = \frac{5}{8}\sin 36^\circ\). Thus \(B = \arcsin(\frac{5}{8}\sin 36^\circ)\).

Step 2: Find Angle C

Since the sum of the internal angles of a triangle is \(180^{\circ}\), we have \(C = 180^{\circ} - A - B\). Thus, \(C = 180^{\circ} - 36^{\circ} - B\).

Step 3: Find Side c

Finally, applying the Law of Sines once more, we can determine the length of side \(c\). From \(\sin C = \frac{c}{a}\), we can isolate \(c\) to get \(c = a\cdot\sin C = 8\cdot\sin C\).

Key Concepts

Triangle Solving

Solving a triangle involves finding unknown sides and angles using known information. In this exercise, we are given the measure of one angle, \( A = 36^{\circ} \), and two sides, \( a = 8 \) and \( b = 5 \). To solve the triangle, we follow a systematic approach using trigonometric formulas such as the Law of Sines. Here, our goal is to find the remaining angle \( B \), the third angle \( C \), and the unknown side \( c \).
This process demonstrates the beauty of triangle solving, where limited known values unveil other hidden properties of the triangle. It reflects the interconnected nature of geometric principles, where one piece of information can lead to the discovery of others.

Angle Calculation

When calculating angles in a triangle, knowing the internal angle sum rule is crucial. The angles inside any triangle always sum up to \( 180^{\circ} \). This fundamental property allows us to determine the unknown angle \( C \) once we have calculated angle \( B \).

In the solution, angle \( B \) is found using the Law of Sines, a powerful trigonometric tool. We have:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} \)

By isolating \( \sin B \) and calculating it, we find \( B = \arcsin(\frac{5}{8}\sin 36^\circ) \). Once \( B \) is known, it's straightforward to find \( C \), computing \( 180^{\circ} - A - B \). This sequential calculation highlights the logical flow needed in angle determination, weaving together known and unknown values effectively.

Trigonometry

Trigonometry is the field of mathematics that helps us solve problems related to angles and sides in triangles. In this case, we use the Law of Sines, an essential formula in trigonometry. This law states:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

This relationship allows us to find unknown sides or angles when certain other elements of the triangle are known.

In our exercise, after determining angles \( B \) and \( C \), we return to the Law of Sines to find the length of side \( c \). Knowing \( \sin C \), we calculate \( c = a \cdot \sin C \). Trigonometry not only helps in solving real-world problems but also provides a systematic approach to understanding geometric relationships, making it a cornerstone in the study of geometry.