Problem 6 - Trigonometry

Question

For each of the following triangles, solve for \(B\) and use the results to explain why the triangle has the given number of solutions. \(A=30^{\circ}, b=40 \mathrm{ft}, a=15 \mathrm{ft}\); no solution

Step-by-Step Solution

Answer

The triangle has no solution because \(\sin B = \frac{4}{3}\) is impossible.

Step 1: Understanding Given Information

We are given a triangle where angle \(A = 30^{\circ}\), side \(b = 40 \text{ ft}\), and side \(a = 15 \text{ ft}\). Our task is to determine angle \(B\), and examine the results to conclude why there's no solution.

Step 2: Calculate Possible Angle B Using the Law of Sines

The law of sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Plugging in the given values: \( \frac{15}{\sin 30^{\circ}} = \frac{40}{\sin B} \), which simplifies to \( \frac{15}{0.5} = \frac{40}{\sin B} \), or \( 30 = \frac{40}{\sin B} \).

Step 3: Solve for Sin B

We rearrange the equation to find \( \sin B = \frac{40}{30} \). Simplify to \( \sin B = \frac{4}{3} \).

Step 4: Check Feasibility of Sin B

Note that \( \sin B = \frac{4}{3} \) is not possible since the sine of an angle must be between \(-1\) and \(1\). Since \(\sin B\) is greater than \(1\), angle \(B\) cannot exist.

Step 5: Conclude Number of Solutions

Since angle \(B\) does not exist, there is no valid triangle configuration with the given side lengths and angle \(A\). Therefore, the problem has no solution.

Key Concepts

Triangle Solutions

When working with triangles, one fundamental task is solving them, which often means finding unknown sides or angles. The Law of Sines is a valuable tool in these situations.
In the exercise scenario, identifying whether a triangle has a solution involves:

• Understanding the given data, which includes sides and angles.
• Applying trigonometric rules, such as the Law of Sines, to find missing angles.
• Assessing the feasibility of these conditions to determine if a solution exists.

The final step is to examine the results you derive from these calculations. In this case, although calculations were performed to find angle \(B\), discovering that \( \sin B = \frac{4}{3} \) acted as a critical factor in concluding that no solutions exist due to trigonometric constraints.

Angle Calculation

In trigonometry, computing angles using given sides and angles forms the essence of solving for part of a triangle. The Law of Sines states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \] This equation aids in determining unknown angles when certain sides and angles are known.
Here, substituting the known values resulted in the expression:
\[ \frac{15}{0.5} = \frac{40}{\sin B} \], which simplifies to \[ 30 = \frac{40}{\sin B} \].
Rearranging gives \( \sin B = \frac{4}{3} \), aiming to solve for angle \(B\).
However, with sin \(B\) being more than 1, it indicated a problematic result, confirming no valid angle could satisfy these conditions.

Trigonometric Feasibility

Understanding the feasibility of triangle solutions requires a grasp of trigonometric principles like the range of sine, cosine, and tangent functions.
For angle \(B\), the calculation \( \sin B = \frac{4}{3} \) doesn’t make sense as sine values are correctly bounded between \(-1\) and \(1\).
This reflects an inconsistency, as real numbers cannot produce values beyond these limits for trigonometric functions.
It’s crucial in trigonometry that results align with feasible values to confirm their validity.
In this problem, since \( \sin B \) exceeds the permissible value, it demonstrates that no actual triangle configuration can manifest, rendering the setup devoid of solutions.