Problem 9 - Precalculus with Limits

Question

In Exercises 5-24, use the Law of Sines to solve the triangle.Round your answers to two decimal places. \(A\ =\ 102.4^{\circ}\), \(C\ =\ 16.7^{\circ}\), \(a\ =\ 21.6\)

Step-by-Step Solution

Answer

After calculation: \( B = 60.9^{\circ} \), \( b = 12.05 \) and \( c = 6.13 \)

Step 1: Find Angle B

First, calculate angle B, which can be found by subtracting the known angles A and C from 180. So, \( B = 180 - A - C = 180 - 102.4 - 16.7 = 60.9^{\circ} \)

Step 2: Find Side b

Next, side b can be found by using the Law of Sines which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle. Here, this gives us: \( b = a \cdot \frac{\sin(B)}{\sin(A)} = 21.6 \cdot \frac{\sin(60.9)}{\sin(102.4)} \)

Step 3: Find Side c

Finally, similarly to step 2, side c can be found: \( c = a \cdot \frac{\sin(C)}{\sin(A)} = 21.6 \cdot \frac{\sin(16.7)}{\sin(102.4)} \)

Key Concepts

Triangle Solving

Solving a triangle means finding all its unknown sides and angles based on given values.
For any triangle, the sum of its internal angles is always 180 degrees.
This fundamental property helps us find a missing angle if the other two are known.
In this specific exercise, we know two angles, A and C. Using the property of triangle angle sum, we can quickly calculate the third angle, B.

• Given angles: A = 102.4° and C = 16.7°.
• Calculation: B = 180° - A - C = 60.9°.

Knowing all three angles is crucial because it prepares us to find the missing sides using trigonometric ratios. This strategy is essential not just for this example but for effectively tackling any triangle solving problem.

Trigonometry

Trigonometry is the study of triangles, specifically right and non-right angled triangles.
It involves the relationships between side lengths and angles.
The Law of Sines is a powerful tool in trigonometry.
This law relates the sides and angles of a triangle, allowing us to solve for unknown lengths when angles and an opposite side are known.
For this exercise, we started with a given side, a, and used the Law of Sines to find the unknown sides, b and c.

• The equation for side b: \[ b = a \cdot \frac{\sin(B)}{\sin(A)} \]
• Here, substitute the given side a = 21.6, and angles A and B to find b.
• For side c, use: \[ c = a \cdot \frac{\sin(C)}{\sin(A)} \]
• Again, substitute a and the angles A and C to solve for c.

This approach helps cement how angles and sides interact, a core concept in trigonometry.

Angle Calculation

Calculating angles in a triangle involves both basic arithmetic and understanding triangle properties.
When two angles are given, inducing the third using the angle sum property is straightforward.
Once we know all angles, trigonometric laws like the Law of Sines come into play to determine side lengths.
This dual use of angles—first calculated arithmetically and then applied functionally—is central to solving more complex triangle problems.The calculation involves basic steps:

• Subtract the sum of the known angles from 180° to find the unknown angle.
• Use \( \sin \) functions as part of solving for missing sides.

By using \( \sin \, A \), \( \sin \, B \), and \( \sin \, C \) in our computations, the angles actively lead to the complete solving of the triangle. Understanding this connection greatly enhances problem-solving skills in geometry.