Problem 5 - 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. \(A=150^{\circ}, b=30 \mathrm{ft}, a=10 \mathrm{ft}\); no solution

Step-by-Step Solution

Answer

The triangle has no solution because \( \sin B = 1.5 \) is impossible; hence, no such angle \( B \) exists.

Step 1: Understanding the Law of Sines

We begin by using the Law of Sines, which states \( \frac{a}{\sin A} = \frac{b}{\sin B} \). We will use the given measurements to solve for \( \sin B \).

Step 2: Applying the Law of Sines

Substitute \( a = 10 \), \( A = 150^{\circ} \), and \( b = 30 \) into the equation: \( \frac{10}{\sin 150^{\circ}} = \frac{30}{\sin B} \). Simplifying, we get: \( \frac{10}{0.5} = \frac{30}{\sin B} \).

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

This simplifies to \( 20 = \frac{30}{\sin B} \). Solving for \( \sin B \), we find: \( \sin B = \frac{30}{20} = 1.5 \).

Step 4: Analyzing \( \sin B \) Value

\( \sin B = 1.5 \) is not possible, because the sine of an angle must be between -1 and 1. This indicates that there is no angle \( B \) that satisfies these conditions.

Step 5: Conclusion on Number of Solutions

Since there is no valid \( B \) that can be calculated from \( \sin B \ = 1.5 \), this means that the triangle does not exist. Therefore, the triangle has no solution.

Key Concepts

Triangle Solutions

When solving a triangle, especially a non-right triangle, the Law of Sines is often a handy tool. This rule lets us calculate unknown angles and sides when given certain measurements. For any given triangle, knowing any three elements (including at least one side) allows you to find the others. The equation \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) is the backbone of this method. With this equation, if two sides and an angle are known, you can solve for a missing part. In our case, parameters included \( A = 150^{\circ} \), \( a = 10 \text{ ft} \), and \( b = 30 \text{ ft} \). On plugging these into the formula, an attempt to discover the missing angle \( B \) was made. Proper solutions hinge on all parts fitting within defined mathematical limits, as is evident in the exercise that led to a no solution outcome.

Sine Function Limitations

The sine function, denoted as \( \sin \), describes the ratio of the opposite side to the hypotenuse in a right triangle, and it spans a very specific range in a unit circle. The crucial limit of \( \sin \) values is between -1 and 1. This makes it ideal to always check calculated sine values against these boundaries. In our effort to find \( \sin B \), calculations led us to a value of 1.5. Since this value is outside the permissible range for sine, it raised a red flag. No angle exists where its sine would equal 1.5. The error signals something off in the triangle's expected configuration or dimensions. Ensuring sine values fall within \([-1, 1]\) is a good safeguard in all calculations involving the sine function.

Non-existent Triangles

In geometry, a triangle is said to be non-existent when its parameters contradict fundamental definitions, making it impossible to construct. Using the Law of Sines, solving for \( \sin B \) yielded an invalid result, thus indicating the triangle described was not feasible within the realm of real, constructible figures. Understanding when a solution indicates non-existence helps avoid misinterpretations. Thus, non-existence was verified not just because a side or angle seemed off, but because the computed mathematical parameter (sine of an angle) fell outside valid ranges. In triangle calculations, an outcome suggesting \( \sin X > 1 \) or \( \sin X < -1 \) signifies a necessary re-evaluation, often concluding in a scenario where no such triangle can be drawn or visualized.