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Chapter 6: Problem 45

DISCOVER: Number of Solutions in the Ambiguous Case We have seen that when the Law of sines is used to solve a triangle in the SSA case, there may be two, one, or no solution(s). Sketch triangles like those in Figure 6 to verify the criteria in the table for the number of solutions if you are given \(\angle A\) and sides \(a\) and \(b\) $$\begin{array}{|c|c|} \hline \text { Criterion } & \text { Number of solutions } \\ \hline a \geq b & 1 \\ b>a>b \sin A & 2 \\ a=b \sin A & 1 \\ a

Short Answer

Expert verified
Range: 0 solutions if \( a < 50 \), 1 solution if \( a = 50 \) or \( a \geq 100 \), 2 solutions if \( 50 < a < 100 \).

Step by step solution

01

Identify Known Values

We are given that \( \angle A = 30^\circ \) and side \( b = 100 \). These will be used along with the criteria for deciding the number of solutions.
02

Calculate \( b \sin A \)

To apply the criteria, we need to find \( b \sin A \). Use the equation \( b \sin A = 100 \times \sin 30^\circ = 100 \times 0.5 = 50 \).
03

Apply Solution Criteria

We now analyze each given criterion to determine the range of \( a \):- **Criterion: \( a \geq b \)**: Here, \( a \geq 100 \). This gives 1 solution.- **Criterion: \( b > a > b \sin A \)**: This translates to \( 100 > a > 50 \). This condition provides 2 solutions.- **Criterion: \( a = b \sin A \)**: It results in \( a = 50 \). This gives 1 solution.- **Criterion: \( a < b \sin A \)**: This means \( a < 50 \). It results in no solution.
04

Summarize Results

By consolidating the findings:- If \( a < 50 \), there are 0 solutions.- If \( a = 50 \), there is 1 solution.- If \( 50 < a < 100 \), there are 2 solutions.- If \( a \geq 100 \), there is 1 solution. This matches the criteria exactly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ambiguous Case
The ambiguous case in the Law of Sines arises specifically with the SSA triangle setup. Think about it as a puzzle with some potential surprises. In this situation, you have two sides and a non-included angle of the triangle given.

Here's where the mystery unfolds: these pieces of information can sometimes lead to different possible triangles that satisfy the given conditions. Therefore, using SSA with the Law of Sines can result in more than one solution, and this situation is famously known as the ambiguous case.
  • There could be two triangles that fit the given sides and angle.
  • Possibly, just one triangle might work.
  • Or even, no triangle will satisfy the conditions.
Recognizing this ambiguity is crucial in precalculus trigonometry to predict the number of solutions accurately.
SSA Triangles
SSA triangles refer to a configuration where two sides and an angle not between them are known. This is a specific type of triangle commonly dealt with in trigonometry. The main challenge with SSA triangles is predicting whether you have enough information to form a valid triangle.

When the Law of Sines is applied to solve an SSA triangle, the outcomes can be diverse. This can result from angle-side-side not uniquely determining a triangle like the other setups (e.g., SAS, ASA). Thus, SSA is less straightforward.
  • The known angle is not between the two sides, adding to the uncertainties.
  • Depending on the values, more than one triangle could be possible.
Understanding SSA triangles helps in solving these trigonometric problems efficiently.
Number of Solutions in Triangles
Identifying the number of solutions in triangles, especially with SSA cases, involves analytical thinking. Using the Law of Sines, one assesses the criteria to resolve how many possible triangles correspond to the given data.

For example, given an angle \(\angle A\) and two sides, \(a\) and \(b\), the number of solutions is determined by comparing side lengths and angles:
  • If \(a \geq b\), there is exactly one solution.
  • If \(b > a > b \sin A\), there are two possible solutions.
  • If \(a = b \sin A\), there's exactly one solution.
  • If \(a < b \sin A\), no triangle can form (zero solutions).
Using these criteria allows you to find the range of values for side \(a\) that will result in zero, one, or two solutions.
Trigonometry in Precalculus
Trigonometry in precalculus lays the groundwork for understanding more advanced math by focusing on the properties of triangles and the relationships between their angles and sides.

In the context of precalculus, students explore concepts like the Law of Sines and Cosines, which help solve real-world problems by applying mathematical reasoning to triangle measurements.
  • The Law of Sines is particularly powerful for cases like the ambiguous SSA where direct measurement is less obvious.
  • Understanding trigonometry's applications helps blend geometric reasoning with algebraic methods.
Solidifying these concepts in precalculus prepares students for the intricacies of calculus and other higher-level math courses, where these trigonometric principles continue to play a significant role.

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