Problem 31 - Precalculus with Limits

Question

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=120^{\circ}, \quad a=b=25$$c

Step-by-Step Solution

Answer

The two possible solutions for the triangle are: 1st solution: A = 120 degrees, B = 60 degrees, C = 0 degrees, a = 25, b = 25, c = 0 and 2nd solution: A = 120 degrees, B = 120 degrees, C = -60 degrees (or 300 degrees in terms of positive degrees), a = 25, b = 25 and c approximately equals to 43.30.

Step 1: Identifying Triangle Specifics

In looking at the problem, it appears we have a triangle with sides a and b that are equal to each other (both 25 units long), and an angle A that measures 120 degrees. This is a SAS (Side-Angle-Side) scenario.

Step 2: Applying the Law of Sines

The Law of Sines allows us to figure out the proportions of the triangle, using the formula \( \frac{a}{\sin{A}} = \frac{b}{\sin{B}} \). Since a and b are equal, the equation simplifies down to \( \sin{B} = \sin{A} \). Substituting the given values, we get \( \sin{B} = \sin{120} \). Solve for B.

Step 3: Calculate Angle B

The value of \( \sin{120} \) is approximately 0.866, meaning that B equals to \(\sin^{-1}(0.866)\) and this results in two possible values for B: B = 60 and B = 120 degrees (supplement of 60 degrees in the unit circle). We must keep both of these values in mind as we progress to the next step.

Step 4: Applying the Triangle's Angle Sum

Every triangle's angles add up to 180 degrees. So the missing angle C could be calculated by subtracting A and B from 180. For both cases of B equals to 60 and 120 degrees, calculate angle C.

Step 5: Applying the Law of Sines for side c

Use the Law of Sines again to compute the unknown side length c, using the formula \( \frac{c}{\sin{C}} = \frac{a}{\sin{A}} \). Calculate c for both cases.

Key Concepts

SAS (Side-Angle-Side)

Understanding the SAS (Side-Angle-Side) condition is crucial for solving many types of triangle problems. It refers to when we know two sides of a triangle and the angle between them. In trigonometry, this sets up a scenario where we can apply certain rules and formulas to find the remaining sides and angles.

For instance, if we know sides 'a' and 'b' and the angle 'A' sandwiched between them, as in the exercise, we're in a good position to start solving for the rest of the triangle's measurements. The key here is that the given angle must be between the known sides for it to be considered a SAS scenario. This information allows us to use the Law of Sines or potentially the Law of Cosines (not needed here), depending on what we are solving for.

Trigonometric Functions

Trigonometric functions are the foundation of solving triangles in trigonometry. These functions - sine (sin), cosine (cos), and tangent (tan), and their inverses - relate the angles of a triangle to the lengths of its sides.

When we are given an angle, like angle 'A' at 120 degrees in the exercise, we use the sine function to find the ratio of the opposite side 'a' to the hypotenuse in a right triangle. This is the 'sin' value. However, in non-right triangles, these functions can still be applied by constructing an imaginary right triangle or by using the extended Law of Sines or Law of Cosines formulas.

The Power of Sine

The sine function is particularly powerful because it allows us to calculate an unknown side or angle when dealing with a pair of opposite sides and angles in the triangle, as demonstrated in our given exercise.

Triangle Angle Sum Theorem

The triangle angle sum theorem is a fundamental principle in geometry that states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for solving triangles, as it provides a way to find a missing angle once two angles are known.

Using the triangle angle sum theorem, if you know two angles, you can always find the third by subtracting the sum of the known angles from 180 degrees. It's a simple but powerful tool that appears frequently in geometry problems. In the context of the exercise, once the angle 'B' was determined, the triangle angle sum theorem helped to easily figure out the measure of angle 'C' by subtraction from 180 degrees.

Ambiguous Case

The ambiguous case arises when we deal with the Law of Sines and have two possible solutions for a triangle. This typically occurs in situations involving non-right triangles when using SSA (Side-Side-Angle) information, where we know two sides and an angle that is not between them.

In our exercise, after calculating the sine value for angle 'B', two possible angles were found. This is because the sine function is positive in both the first and second quadrants, leading to two angle possibilities (B and its supplement). As a result, there could be two different triangles that satisfy the given conditions, hence the ambiguity.

Resolving the Ambiguity

To resolve the ambiguous case, both potential angles are explored to determine if one or two valid triangles exist. By calculating the remaining angle using the triangle angle sum theorem, and then the unknown side with the Law of Sines, we can confirm which cases are actually possible based on the triangle's geometry.