Problem 19 - Algebra and Trigonometry

Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. $$ a=10, c=8.9, A=63^{\circ} $$

Step-by-Step Solution

Answer

The given side lengths and angle form one triangle. The missing values are B ~ 63 degrees, C ~ 54 degrees, and b ~ 8.6 units.

Step 1: Verify if a Triangle Can Form

The first important step is to determine whether the given measurements form a valid triangle or not. This can be done by checking if the side opposite to the given angle (c in this case) is longer than the side 'a' multiplied by the sine of angle A. Mathematically, this can be written as c > a*sin(A).

Step 2: Determine Number of Possible Triangles

If c does indeed exceed a*sin(A), then there exist two possible triangles. If the equality is strict, then there exists one triangle. If c < a*sin(A), no triangles exist. The first step should have provided clarity on the number of possible triangles.

Step 3: Compute Remaining Angle and Sides

This step is carried out only if there exists at least one valid triangle. Start by using the Law of Sines to calculate Angle B: sin(B) = (b * sin(A)) / a. Next, use the fact that the sum of angles in a triangle always equals 180 degrees to calculate Angle C: C = 180 - A - B. Round the values of B and C to the nearest degree. Lastly, use the Law of Sines to find side b: b = a * sin(B) / sin(A). Round the value of b to the nearest tenth.

Key Concepts

Law of Sines

Understanding the Law of Sines is crucial in solving SSA triangles, as it provides a relationship between the lengths of sides and the sines of the opposite angles in any given triangle.

The Law of Sines states that for any triangle with sides a, b, and c, and angles A, B, and C opposite those sides, respectively, the following proportion holds: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \.\] This formula is particularly useful in scenarios where we know two sides and a non-included angle (SSA) and want to solve for unknown parts of the triangle.

For example, given sides a and c, and angle A in our exercise, we could apply the Law of Sines to find the angle B as follows: \[ \sin(B) = \frac{b \cdot \sin(A)}{a} \.\] After determining B, we can calculate the third angle, C, and the remaining side, b, using this law. This method is effective in finding missing angles and sides, provided that we have the necessary information to apply the formula correctly.

Triangle Ambiguity

Triangle ambiguity arises in the context of SSA problems, where it's often uncertain if given side lengths and an angle will result in one triangle, two different triangles, or no triangle at all. This ambiguity is a critical concept to understand as it can significantly affect the solutions of triangle problems.

When presented with two sides and an angle that is not included between the sides (SSA condition), the following can occur:

• If the given side opposite the known angle is the longest side, only one triangle is possible.
• If the given side is shorter than the other side but longer than the altitude from the angle, two different triangles can be formed.
• If the given side is shorter than the altitude from the angle, no triangle can exist.

By comprehending the conditions that lead to triangle ambiguity, students are better equipped to proceed with solving SSA triangle problems accurately.

SSA Triangle Problem

The SSA triangle problem is the challenge of determining whether the given measurements for two sides and an angle can form a triangle, and if so, how many distinct solutions there are.

Following the example from our exercise, we're given sides a and c, and angle A. The first step in addressing an SSA problem is to check whether a valid triangle can form using the inequality \[ c > a \cdot \sin(A) \.\] If this condition is met, the next step is to explore the number of possible triangles, which hinges on whether side c is greater than, equal to, or less than the product of side a and the sine of angle A.

If exactly one triangle can be formed, this typically indicates that the given side opposite the angle is longer than the other side, leading to a unique solution. When the given side is just long enough to meet the tip of the other side when swung like a pendulum, only that position will lead to a singular triangle. However, if the side is shorter and allows for two positions when swung, two distinct triangles can be formed. This problem exemplifies the importance of careful analysis and understanding when working through SSA triangles to avoid incorrect conclusions.