Problem 26 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=110^{\circ}, \quad a=125, \quad b=200$$

Step-by-Step Solution

Answer

A triangle with the given sides and angles cannot be constructed, as the input values do not adhere to the Law of Sines.

Step 1: Identify given sides and angles and label the triangle

The given triangle has an angle A=110 degrees, side a=125 and side b=200.

Step 2: Apply the Law of Sines to find unknown angle

Use the Law of Sines to find the second angle B. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all sides and angles in a given triangle. Therefore, we can write it as: \(\frac{a}{\sin A} = \frac{b}{\sin B}\) Solving this for \(\sin B\) gives: \(\sin B = \frac{b \cdot \sin A}{a}\) Substitute the given values into the equation: \(\sin B = \frac{200 \cdot \sin 110^\circ}{125}\)

Step 3: Calculate the value of sin B

Upon calculating the above expression, we get \(\sin B \approx 1.37\). However, the value of sine of an angle cannot exceed 1, which means that there is a mistake and hence, no solution exists as per this condition.

Step 4: Analyze the inputs

The inputs provided do not adhere to the Law of Sines and hence a triangle with the given sides and angles cannot be constructed. The given values are not geometrically possible, thus there aren't two possible solutions to begin with.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It fundamentally focuses on right-angled triangles but extends its concepts to all triangle types. The core tools of trigonometry include various trigonometric functions, such as sine, cosine, and tangent, which relate angles to the ratios of a triangle’s sides.

These relationships are crucial for solving problems involving angles and distances, whether you are measuring land, engineering structures, or even navigating by stars. One major concept within trigonometry is the Law of Sines, which helps understand and solve triangles that are not necessarily right-angled, allowing for calculation of unknown sides or angles in oblique triangles.

• The sine function is expressed as the ratio of the opposite side length to the hypotenuse in a right triangle.
• The Law of Sines is particularly useful with non-right-angled triangles, offering a solution when two angles and one side are known, or two sides and a non-included angle are known.

Understanding the role of trigonometry helps bridge the gap between abstract math and practical applications.

Solving Triangles

Solving triangles involves finding the unknown sides and angles of a triangle. There are two main laws for solving triangles: the Law of Sines and the Law of Cosines. In this exercise, we apply the Law of Sines, which is used when we are given:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA)

The Law of Sines states that for any triangle with angles A, B, and C, and sides a, b, and c opposite those angles respectively: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
By applying this, we can message out unknown measurements if at least some measurements are known. One must be careful, especially with SSA cases, as they can lead to ambiguous situations, meaning there may be one solution, two solutions, or no solution.

In this specific example, even though we attempted to utilize the Law of Sines, no valid solution could be found. This shows that sometimes the given values do not result in a possible triangle according to the rules of trigonometry. It's important to check the calculations at the end to ensure their viability.

Angle-Side Relationships

Understanding the relationship between angles and sides is key in trigonometry. The angles in a triangle determine how its sides relate to one another. In particular, the Law of Sines helps establish and use these relationships. The law equates the ratio of a side's length to the sine of its opposite angle across the whole triangle. This principle implies several things:

• When given specific side lengths and angles, the triangle's configuration must mathematically adhere to these ratios.
• The triangle inequality theorem should be satisfied; the sum of lengths of any two sides must be greater than the third side.
• If calculated sine values for an angle exceed 1, it indicates an impossible triangle under those given conditions.

The lack of a solution in the original exercise highlights how crucial understanding these relationships is. It signifies that the provided sides and angles could not form a cohesive triangle. This outcome is a great opportunity to revisit core trigonometric concepts and verify inputs before committing to full calculations, ensuring realistic and accurate mathematical modeling.