Problem 37 - Algebra and Trigonometry

Question

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). $$ a=7, \quad c=3, \quad C=12^{\circ} $$

Step-by-Step Solution

Answer

Two triangles can be formed; one with angles \(29.1^\circ, 138.9^\circ\) and side \(b \approx 10.7\), another with angles \(150.9^\circ, 17.1^\circ\) and side \(b \approx 4.24\).

Step 1 - Analyzing the Given Information

We are given two sides and a non-included angle (SSA). In triangle problems, this can potentially lead to no triangle, one triangle, or two triangles. The given information is: - Side a = 7 - Side c = 3 - Angle C = 12°

Step 2 - Determine the Potential Number of Triangles

To determine the number of possible triangles, use the Law of Sines to calculate the height (h) from base c: \[h = c \times \sin(C) = 3 \times \sin(12^\circ) = 3 \times 0.2079 = 0.6237\]. Since h < c and \ a > c \ , there are two potential triangles.

Step 3 - Calculating Angle A for Two Potential Triangles

Use the Law of Sines to find possible values for Angle A: \[ \sin(A) = \frac{a \sin(C)}{c} = \frac{7 \sin(12^\circ)}{3} = \frac{7 \times 0.2079}{3} = 0.4855\]. \ This can have two solutions: 1. \ A = \sin^{-1}(0.4855) \approx 29.1^\circ 2. \ A = 180^\circ - 29.1^\circ = 150.9^\circ.

Step 4 - Using Angle A to Find Remaining Angles and Sides

For the two triangles, solve for Angle B and side b: 1. For \ A = 29.1^\circ: - \ B = 180^\circ - A - C = 180^\circ - 29.1^\circ - 12^\circ = 138.9^\circ - Using Law of Sines, \ b \ can be found: \[ b = \frac{c \sin(B)}{\sin(C)} = \frac{3 \sin(138.9^\circ)}{\sin(12^\circ)} = \frac{3 \times 0.7431}{0.2079} \approx 10.7\]2. For \ A = 150.9^\circ: - \ B = 180^\circ - A - C = 180^\circ - 150.9^\circ - 12^\circ = 17.1^\circ - Side b: \[ b = \frac{c \sin(B)}{\sin(C)} = \frac{3 \sin(17.1^\circ)}{\sin(12^\circ)} = \frac{3 \times 0.294} {0.2079} \approx 4.24\]

Key Concepts

triangle construction

Triangle construction involves determining the dimensions and angles of a triangle based on given information. In this particular problem, we are given two sides (a and c) and a non-included angle (C). This type of information falls under the 'SSA condition,' which stands for Side-Side-Angle. To construct a triangle, we typically use trigonometric principles to determine the other sides and angles. Our goal is to determine the number of possible triangles that can be formed with the provided data and to calculate their respective dimensions.

ambiguous case

The ambiguous case (SSA condition) occurs when two sides and a non-included angle of a triangle are given, leading to the possibility of multiple solutions or no solution at all. This case is tricky because:

• One triangle might fit the given criteria.
• Two different triangles might fit the criteria.
• No triangle at all could fit.

To resolve the ambiguous case, we use the Law of Sines to check if there are possible triangles and find angle and side values for the potential triangles.

trigonometric functions

Trigonometric functions are crucial in solving triangles, especially in the context of the Law of Sines. For a triangle with sides a, b, c and angles A, B, C opposite those sides, the Law of Sines states:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
Trigonometric functions, like sine (\

SSA condition

The SSA condition is when we are given two sides and a non-included angle in a triangle. With SSA, the Law of Sines is used to determine:

• Whether a valid triangle can be formed with the given information.
• If one, two, or no triangles exist.

The process involves:

• Calculating the height (h) using \( h = c \times \sin(C) \).
• Comparing the height to the given sides.
• Using the Law of Sines to find potential angle values and verifying the feasibility of the triangle(s).

In this exercise, we determined there are two potential triangles because the calculated height is less than side c and side a is greater than side c, indicating multiple triangle possibilities.