Problem 12 - Trigonometry

Question

Find all solutions to each of the following triangles: \(A=43^{\circ}, a=31 \mathrm{ft}, b=37 \mathrm{ft}\)

Step-by-Step Solution

Answer

The triangle has two possible solutions due to the ambiguous case for angle B.

Step 1: Understand the Problem

We need to find the missing side and angles of a triangle given that \( A = 43^{\circ} \), \( a = 31 \text{ ft} \), and \( b = 37 \text{ ft} \). This calls for the use of the Law of Sines because we have an angle-side pair.

Step 2: Apply Law of Sines

Using the Law of Sines: \( \frac{\sin A}{a} = \frac{\sin B}{b} \). Plug in the known values: \( \frac{\sin 43^{\circ}}{31} = \frac{\sin B}{37} \).

Step 3: Solve for \( \sin B \)

Rearrange the equation to find \( \sin B \): \( \sin B = \frac{37 \cdot \sin 43^{\circ}}{31} \). Calculate \( \sin 43^{\circ} \) and find the product and division.

Step 4: Calculate Angle \( B \)

Using the value of \( \sin B \) calculated in Step 3, use the inverse sine function to find \( B \). Check if \( \sin B \) leads to a valid angle or more than one possible angle (Ambiguous Case).

Step 5: Find Angle \( C \)

Calculate angle \( C \) by using the sum of angles in a triangle: \( C = 180^{\circ} - A - B \).

Step 6: Use Law of Sines to Find \( c \)

Now that all angles are known, use the Law of Sines again to find side \( c \): \( \frac{c}{\sin C} = \frac{a}{\sin A} \). Rearrange and plug in the known values to find \( c \).

Step 7: Verify Solution

Check the calculated angles and side to ensure they satisfy the triangle properties and match the given conditions.

Key Concepts

Solving Triangles

Solving a triangle means finding all the unknown sides and angles of a triangle. In problems like this one involving the Law of Sines, you are typically given either:

• two angles and one side (AAS or ASA), or
• two sides and a non-included angle (SSA).

To solve a triangle using these methods, follow these straightforward steps:1. **Identify the Given Information:** Determine which sides and angles you know. In our triangle, we know angle \( A = 43^{\circ} \), side \( a = 31 \text{ ft} \), and side \( b = 37 \text{ ft} \). This is an SSA case.
2. **Apply the Law of Sines:** This formula connects angles and sides of the triangle: \[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \] Start with the known angle-side pair and the sides given to find missing angles.
3. **Find the Unknowns:** Use calculated angles to determine the unknown side using the Law of Sines again.
Solving triangles step-by-step helps ensure accuracy and clarity in your answers.

Angle-Side Pair

An angle-side pair in trigonometry involves a known angle and the side opposite to it. Understanding this concept is crucial when applying the Law of Sines:

• Having an angle-side pair means you know both the measure of the angle and the length of the side directly opposite it.
• In this exercise, our known angle-side pair is angle \( A = 43^{\circ} \) and side \( a = 31 \text{ ft} \).

This pair serves as an anchor - it helps set up the primary ratio in the Law of Sines formula. Knowing these numbers allows you to calculate the sine of the other angles when you have other side lengths available. When paired correctly, you can establish the equation as:

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

This ratio is then rearranged to solve for the missing sine function, leading you closer to the full solution of the triangle. Understanding and using angle-side pairs correctly is key to unlocking unknowns in triangle problems.

Ambiguous Case

The ambiguous case arises in the SSA (Side-Side-Angle) scenario in trigonometry. In simple terms, this means there can be more than one possible triangle that fits the given information.

• This case appears because the calculated angle using the inverse sine function could lead to two possible angles.
• To explain, if \( \sin B \) gave you 30 degrees, then \( B \) could also be \(180^{\circ} - 30^{\circ} = 150^{\circ} \), because sine values repeat every 180 degrees.

To handle the ambiguous case:- First, calculate the initial possible angle \( B \) using the inverse sine of the calculated value.- Consider its supplement (180 degrees minus the angle) as another possibility.- Check both angles in the sum of angles in the triangle (which must equal 180 degrees) to see if each results in a feasible triangle.
Handling the ambiguous case by examining possible triangles ensures you don't miss a valid solution. This careful analysis helps you determine if there's indeed one or two solutions based on the initial given triangle data.