Problem 32 - 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). $$ b=2, \quad c=3, \quad B=40^{\circ} $$

Step-by-Step Solution

Answer

There are two possible triangles. The first triangle has angles 40°, 74.06°, 65.94° and sides a ≈ 2.98, b = 2, c = 3. The second triangle has angles 40°, 105.94°, 34.06° and sides a ≈ 1.82, b = 2, c = 3.

Step 1 - Determine the possible scenarios using Law of Sines

First, use the Law of Sines to find the possible values for angle C. The Law of Sines states that \[ \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \].Plug in the known values:\[ \frac{2}{\sin(40^{\circ})} = \frac{3}{\sin(C)} \].

Step 2 - Solve for \( \sin(C) \)

Rearrange the equation to solve for \( \sin(C) \):\[ \sin(C) = \frac{3 \cdot \sin(40^{\circ})}{2} \].Calculate \( \sin(40^{\circ}) \) and solve:\[ \sin(C) = \frac{3 \cdot 0.6428}{2} = 0.9642 \].

Step 3 - Assess the values for \( C \)

Since \( \sin(C) \leq 1 \), we need to determine whether this value of \( \sin(C) \) yields one or two possible angles. Using the inverse sine function,\[ C = \sin^{-1}(0.9642) \]. This gives:\[ C \approx 74.06^{\circ} \]. Also, if \( C = 105.94^{\circ} \) (since \( \sin(180^{\circ} - x) = \sin(x) \)).

Step 4 - Determine possible triangles

Check if the two possible values of \( C \) lead to valid triangles. Use the triangle angle sum property, \[ A + B + C = 180^{\circ} \].1. If \( C = 74.06^{\circ} \):\[ A = 180^{\circ} - 40^{\circ} - 74.06^{\circ} = 65.94^{\circ} \]. This yields a valid triangle.2. If \( C = 105.94^{\circ} \):\[ A = 180^{\circ} - 40^{\circ} - 105.94^{\circ} = 34.06^{\circ} \]. This also yields a valid triangle.

Step 5 - Solve the triangles

1. For \( C = 74.06^{\circ} \) and \( A = 65.94^{\circ} \), use the Law of Sines to find side \( a \):\[ \frac{a}{\sin(65.94^{\circ})} = \frac{2}{\sin(40^{\circ})} \] Thus,\[ a = \frac{2 \cdot \sin(65.94^{\circ})}{\sin(40^{\circ})} \approx 2.98 \].2. For \( C = 105.94^{\circ} \) and \( A = 34.06^{\circ} \), use the Law of Sines to find side \( a \):\[ \frac{a}{\sin(34.06^{\circ})} = \frac{2}{\sin(40^{\circ})} \] Thus,\[ a = \frac{2 \cdot \sin(34.06^{\circ})}{\sin(40^{\circ})} \approx 1.82 \].

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. One of the foundational concepts in trigonometry is the use of trigonometric functions such as sine, cosine, and tangent to relate the angles and sides of a triangle.
In this exercise, we specifically use the Law of Sines, which is expressed as: \[ \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)} \] This law helps us find unknown angles or sides in a triangle when certain other measurements are known.
By using the Law of Sines, we can solve complex problems and determine whether the given values result in a single triangle, multiple triangles, or no triangle at all.

Triangle Properties

Triangles are simple, yet fundamental geometrical shapes with three sides and three angles. There are several important properties to consider:

• The angles add up to 180 degrees.
• Any side must be shorter than the sum of the other two sides.

The given problem involves a triangle where two sides (b = 2 and c = 3) and an angle (B = 40°) are known.
These properties guide us to use the Law of Sines to find unknown sides or angles, ensuring that the solution meets the criteria for valid triangles.

Angle Sum Property

The angle sum property of a triangle states that the sum of the interior angles is always 180 degrees. This fundamental rule helps us verify the validity of a triangle.
After finding the possible values for angle C using the Law of Sines, we use the angle sum property to find the remaining angle A:

• If C = 74.06°, then A = 180° - 40° - 74.06° = 65.94°.
• If C = 105.94°, then A = 180° - 40° - 105.94° = 34.06°.

This property ensures that the angles calculated are valid and form a triangle.

Inverse Trigonometric Functions

Inverse trigonometric functions help us find the angles when we know the value of a trigonometric ratio.
In this problem, we need the inverse sine function to find the angle C given that \[ \text{sin}(C) = 0.9642 \] By using the inverse sine function (sin⁻¹), we find:

• C ≈ 74.06°
• Also, C can be 180° - 74.06° = 105.94°

These two angles demonstrate how the inverse trigonometric functions provide possible values that need further verification for valid triangle determination.

Triangle Determination

Determining if the given data results in a valid triangle involves several steps:

• Use the Law of Sines to find possible values.
• Check with the angle sum property to ensure the angles add up to 180 degrees.
• Verify that the sides conform to triangle inequality rules.

In the given exercise, using the sides b = 2, c = 3, and angle B = 40°, we found two sets of possible angles and sides. Both are valid, leading us to conclude the given data results in two distinct triangles:

• Triangle 1: C ≈ 74.06°, A ≈ 65.94°, a ≈ 2.98
• Triangle 2: C ≈ 105.94°, A ≈ 34.06°, a ≈ 1.82

These steps show how to systematically determine the validity and count of triangles.