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Chapter 6: Problem 2

The four cases in which we can solve a triangle are $$ASA SSA SAS SSS$$ (a) In which of these cases can we use the Law of sines to solve the triangle? (b) Which of the cases listed can lead to more than one solution (the ambiguous case)?

Short Answer

Expert verified
(a) ASA, SSA can use the Law of Sines. (b) SSA case is ambiguous.

Step by step solution

01

Understanding Triangle Cases

The triangle cases mentioned are different scenarios based on the values given (angles or sides). ASA (Angle-Side-Angle) means two angles and the included side are known. SSA (Side-Side-Angle) means two sides and a non-included angle are given. SAS (Side-Angle-Side) indicates two sides and the included angle are known. SSS (Side-Side-Side) means all three sides are given.
02

Identifying Law of Sines Case

The Law of Sines is most effective when dealing with cases that involve angles and their opposite sides. Thus, it can be used for ASA and SSA cases because you can set up proportions that relate the elements of the triangle using the sine function.
03

Identifying the Ambiguous Case

The ambiguous case occurs in the SSA scenario because given two sides and a non-included angle, there might be two different triangles that satisfy these conditions. This is due to the nature of solving angle-side-side configurations with the Sine function, which can yield more than one angle measure.

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Key Concepts

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

Triangle Cases
Triangles can be solved using several cases depending on the information available in terms of sides and angles. Each case is a unique scenario that requires specific approaches to solve:
  • ASA (Angle-Side-Angle): Two angles and the side between them are given.
  • SSA (Side-Side-Angle): Two sides and an angle that is not between them are given.
  • SAS (Side-Angle-Side): Two sides and the angle between them are given.
  • SSS (Side-Side-Side): All three sides are given.
Understanding these cases is essential for applying the right mathematical tools, like trigonometric rules, to find unknown angles or sides of the triangle.
ASA Triangle
The ASA triangle is one where you know two angles and the side included between them. This arrangement is particularly suited for the Law of Sines. By knowing two angles, you can easily calculate the third angle because the sum of angles in a triangle is always 180 degrees:\[ A + B + C = 180^\circ \]Once all angles are known, the Law of Sines can be applied to find the unknown sides. The Law of Sines states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This relationship between the sides and their opposite angles helps calculate the remaining unknown side when one of the sides is known.
SSA Triangle
In an SSA triangle, two sides and a non-included angle are given. This scenario is known for its uniqueness in the sense that it might lead to different triangle solutions, making it an ambiguous case. While the Law of Sines can be used here:\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]There can sometimes be two possible solutions for the angle, leading to two possible triangles. The challenge arises from the fact that for a given SSA setup, the sine of an angle lacks uniqueness, meaning:
  • If the angle calculated is acute (< 90 degrees), there might be another valid obtuse angle that also satisfies the condition.
  • This potential for multiple interpretations is why SSA is sometimes called the "ambiguous case."
Ambiguous Case
The ambiguous case is a situation that arises with the SSA configuration where a given set of data might correspond to two different triangles. This occurs because the sine function can provide two different angle measures that satisfy the equation.
  • When calculating the angle opposite a given side using the Law of Sines with SSA data, you may encounter situations where the angle can be both acute and obtuse.
  • This means that two valid triangles can be formed with the provided data. One triangle would have an acute angle, and the other would have an obtuse angle.
  • This is why the SSA is occasionally tricky and requires additional checks or confirmations to ensure all possible solutions are identified.
Understanding the ambiguity in SSA cases deepens your grasp of triangle solving and the potential challenges involved.

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Most popular questions from this chapter

Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere whose center \(D\) lies on the line passing through the centers of the bubbles (see the figure). Also, \(\angle A C B\) and \(\angle A C D\) each have measure \(60^{\circ} .\) (a) Show that the radius \(r\) of the common face is given by $$ r=\frac{a b}{a-b} $$ [Hint: Use the Law of Sines together with the fact that an angle \(\theta\) and its supplement \(180^{\circ}-\theta\) have the same sine.] (b) Find the radius of the common face if the radii of the bubbles are \(4 \mathrm{cm}\) and \(3 \mathrm{cm} .\) (c) What shape does the common face take if the two bubbles have equal radii? (Image can't copy)

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