Problem 31 - 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). $$ a=7, \quad b=14, \quad A=30^{\circ} $$

Step-by-Step Solution

Answer

One triangle exists with angles \(30^{\text{o}}, 90^{\text{o}},\) and \(60^{\text{o}}\), and sides \(7, 14,\) and \(7\sqrt{3}\).

Step 1: Identify the given information

The given information includes two sides and one angle: \ \( a = 7 \), \( b = 14 \), and \( A = 30^{\text{o}} \).

Step 2: Use the Law of Sines

Using the Law of Sines: \[ \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} \] \ Calculate \(\text{sin}(B)\) using the given values: \ \[ \text{sin}(B) = \frac{b \times \text{sin}(A)}{a} = \frac{14 \times \text{sin}(30^{\text{o}})}{7} \]\ \(\text{sin}(30^{\text{o}}) = 0.5\), so: \ \[ \text{sin}(B) = \frac{14 \times 0.5}{7} = 1 \]

Step 3: Analyze the value of \(\text{sin}(B)\)

Since \(\text{sin}(B) = 1\), \(B\) must be \(90^{\text{o}}\). Therefore, there can only be one triangle because \(\text{sin}(B) = 1\) does not permit any other angles.

Step 4: Calculate the remaining angle \(C\)

\(A\) and \(B\) are known, so \(C\) can be calculated: \ \[ C = 180^{\text{o}} - A - B = 180^{\text{o}} - 30^{\text{o}} - 90^{\text{o}} = 60^{\text{o}} \]

Step 5: Solve for side \(c\) using the Law of Sines

By the Law of Sines: \[ \frac{c}{\text{sin}(C)} = \frac{a}{\text{sin}(A)} \] \ \[ c = \frac{a \times \text{sin}(C)}{\text{sin}(A)} = \frac{7 \times \text{sin}(60^{\text{o}})}{\text{sin}(30^{\text{o}})} \]\ \(\text{sin}(60^{\text{o}}) = \sqrt{3}/2\), so: \ \[ c = \frac{7 \times \sqrt{3}/2}{0.5} = 7 \sqrt{3} \]

Key Concepts

Triangle Determination

To determine the type and number of triangles that fit given measurements, we can use the Law of Sines. This law helps assess whether the given sides and angles can form a triangle, and if more than one triangle is possible. For this problem, we have the following data: side 'a' = 7, side 'b' = 14, and angle A = 30°. Given two sides and an opposite angle, we can start by evaluating the possible scenarios.

Trigonometric Identities

Trigonometric identities are useful tools in solving problems involving triangles. The Law of Sines is one such identity, which states: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] Using this, we solve for the unknowns. Here, we first find sin(B), then determine angle B, and finally solve for side c. For angle A = 30°, knowing that \(\sin(30°) = 0.5\), helps a lot in our calculations.

Angle-Side Relationships

The relationships between angles and sides in a triangle are crucial in solving it. From the earlier calculation, \(\sin(B) = 1\), indicating that B must be 90°. With A = 30° and B = 90°, we can easily find the third angle, \(C\), using the formula: \[ C = 180° - A - B \] This yields C = 60°. These relationships help maintain the consistency of the triangle's properties.

Solving Triangles

Solving a triangle involves finding all its sides and angles using given information. In this exercise, we calculate side c using the Law of Sines: \[ \frac{c}{\sin(C)} = \frac{a}{\sin(A)} \] Given \(C = 60°\) and \(\sin(60°)= \sqrt{3}/2\), we determine: \[ c = \frac{7 \times \sqrt{3}/2}{0.5} = 7 \sqrt{3} \] Thus, the triangles' sides and angles are completely determined, solving the problem fully and showing that only one triangle conforms to the given measures.