Problem 21 - Algebra and Trigonometry

Question

\(19-28\) . Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=20, \quad c=45, \quad \angle A=125^{\circ} $$

Step-by-Step Solution

Answer

No triangle satisfies these conditions as \( \sin C > 1 \).

Step 1: Understanding the Law of Sines

The Law of Sines relates the sides and angles of a triangle. The formula is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We will use this formula to find the unknowns of the triangle given \( a = 20, c = 45, \) and \( \angle A = 125^{\circ} \).

Step 2: Calculate \( \angle C \)

The sum of angles in a triangle is always \( 180^{\circ} \). Given \( \angle A = 125^{\circ} \), we can find \( \angle C \) by first assuming \( \angle C < 55^{\circ} \) (since \( \angle C = 180^{\circ} - \angle A - \angle B \)). However, without \( \angle B \), we can deduce \( \angle C \) after further analysis.

Step 3: Use Law of Sines for \( \angle C \)

Using the Law of Sines, \( \frac{c}{\sin C} = \frac{a}{\sin A} \). Plugging in the known values, \( \frac{45}{\sin C} = \frac{20}{\sin 125^{\circ}} \). Solving for \( \sin C \), we find \( \sin C = \frac{45 \cdot \sin 125^{\circ}}{20} \).

Step 4: Calculate \( \sin C \)

Calculate \( \sin 125^{\circ} \), which is approximately \( 0.8192 \). Thus, \( \sin C = \frac{45 \cdot 0.8192}{20} \approx 1.8432 \). Since \( \sin C \) cannot exceed 1, this implies there's no possible triangle with these measurements.

Key Concepts

Triangle Solving

When tackling the problem of triangle solving, especially using trigonometric concepts like the Law of Sines, it's important to aim at finding all missing sides and angles of a triangle. Each triangle problem presents a unique set of knowns and unknowns. The challenge is to determine the unknowns based on given data.

For a triangle, knowing at least one side and the angles allows us to find the other measurements. In the given exercise, we know side length \(a\), \(c\), and angle \(A\). Using these knowns, other parameters can be found using relationships like the Law of Sines. However, it is crucial to ensure the consistency of triangle conditions, as they may not yield valid geometric triangles if parameters overlap or exceed plausible triangular values.

Trigonometric Functions

Trigonometric functions are central to solving triangles. In particular, functions like sine, cosine, and tangent connect angles to side lengths of right-angled triangles. However, they are extensively utilized in non-right triangles using identities like the Law of Sines and the Law of Cosines.

The Law of Sines’ equation \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) offers a trigonometric method to derive unknown values. In this specific exercise, calculating \(\sin C\) from known values sets the path for determining if the triangle is possible with given dimensions. In such contexts, trigonometric values beyond the physical range, such as results greater than 1 for sine, imply impossibility, flagging the need for reconsideration.

Angle Sum Property

The angle sum property is a fundamental rule in triangle geometry stating that the sum of internal angles in a triangle is always \(180^\circ\). This property serves as a vital check during triangle solving.

In this exercise, gaining insight into angle relationships is vital. With given \(\angle A = 125^\circ\), possible values for \(\angle B\) and \(\angle C\) must be such that they satisfy \(\angle A + \angle B + \angle C = 180^\circ\). Thus, without even knowing \(\angle B\) initially, establishing a potential range for \(\angle C\) helps determine feasibility. If calculations lead to contradictions with this sum property, it confirms that no viable triangle can be formed with the initial set of known measurements.