Chapter 9: Problem 35
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=8, \quad c=3, \quad C=125^{\circ} $$
Short Answer
Expert verified
No triangle can be formed.
Step by step solution
01
Understand the Given Information
Two sides and an angle are given: Side a = 8, Side c = 3, Angle C = 125°. This is a case of SSA (Side-Side-Angle).
02
Check for Possible Triangles
In an SSA condition, check if the given angle C is obtuse or acute. Since C = 125°, it is obtuse. For an obtuse angle, a must be greater than c to form a triangle.
03
Verify the Condition a > c
Compare side a and side c: a = 8 and c = 3. Since 8 > 3, there will be one valid triangle.
04
Use the Law of Sines
Apply the Law of Sines to find another angle. Use the formula: \ \ \( \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \) Substitute the known values: \ \ \( \frac{8}{\sin(A)} = \frac{3}{\sin(125^{\circ})} \) \ Rearrange to solve for \( \sin(A) \): \( \sin(A) = \frac{8 \cdot \sin(125^{\circ})}{3} \)
05
Calculate \( \sin(125^{\circ}) \)
Find the sine of 125°: \( \sin(125^{\circ}) = \sin(180^{\circ} - 55^{\circ}) = \sin(55^{\circ}) \approx 0.8192 \)
06
Solve for \( \sin(A) \)
Plug the value of \( \sin(125^{\circ}) \) into the equation: \( \sin(A) = \frac{8 \cdot 0.8192}{3} \approx 2.184 \) Notice that \( \sin(A) \) cannot be greater than 1. Therefore, \( \sin(A) \) being greater than 1 signifies there is an error, making it impossible to form a triangle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Law of Sines
The Law of Sines is a fundamental concept in trigonometry that helps solve triangles. It relates the angles of a triangle to the lengths of its sides. According to the Law of Sines, for any triangle with sides a, b, and c and opposite angles A, B, and C, we have: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] This ratio is particularly useful when you know two sides and one non-included angle (SSA condition), as it helps you find the unknown angles or sides. In our problem, we utilized the Law of Sines to establish: \[ \frac{8}{\sin(A)} = \frac{3}{\sin(125^{\circ})} \] We then solve for \( \sin(A)\) by isolating it and substituting known values. This makes the Law of Sines a powerful tool for solving complex triangle problems.
SSA Ambiguity
The SSA condition stands for 'Side-Side-Angle' and is notorious in trigonometry for leading to ambiguous cases. This is because, given two sides and a non-included angle, it's often unclear if you can form 0, 1, or 2 triangles. There are a few steps you can follow to resolve this ambiguity:
- First, check if the given angle is acute or obtuse.
- For an acute angle, multiple scenarios like one, two, or no triangles may arise based on the sides' lengths.
- For an obtuse angle, as in our example (125°), the third side must strictly be greater than the given shorter side to form one triangle.
Triangle Validity
Confirming the validity of a triangle is crucial in trigonometry. For a triangle to be valid, the sum of any two sides must be greater than the third side. Furthermore, the angles must sum up to 180°. In the context of the SSA problem:
- Check if \(A + B + C = 180^{\circ} \).
- Ensure the calculated sine values are between -1 and 1 since the sine of an angle can never be more than 1.