Problem 22 - Precalculus Mathematics for Calculus

Question

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$b=45, \quad c=42, \quad \angle C=38^{\circ}$$

Step-by-Step Solution

Answer

Two triangles: (1) \( A=96.34^{\circ}, B=45.66^{\circ}, a=61.9 \); (2) \( A=7.66^{\circ}, B=134.34^{\circ}, a=6.01 \).

Step 1: Recall Law of Sines

The Law of Sines states that for any triangle with sides \( a, b, c \) and opposite angles \( A, B, C \), the ratio \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) holds true. We will use this to find the missing angles and sides.

Step 2: Apply Law of Sines to Find Angle B

Given \( b = 45 \), \( c = 42 \), and \( \angle C = 38^{\circ} \), we use the Law of Sines: \( \frac{b}{\sin B} = \frac{c}{\sin C} \). This gives us \( \frac{45}{\sin B} = \frac{42}{\sin 38^{\circ}} \). Solving for \( \sin B \), we find \( \sin B = \frac{45 \sin 38^{\circ}}{42} \). Calculate this to find \( \sin B \approx 0.716 \).

Step 3: Determine Possible Values for Angle B

Since \( \sin B \approx 0.716 \), \( \angle B \) can be either \( \approx 45.66^{\circ} \) or \( \approx 134.34^{\circ} \) due to the sine function's property (\( \sin(180^{\circ} - x) = \sin(x) \)). We must check both scenarios.

Step 4: Evaluate the First Triangle with \( \angle B \approx 45.66^{\circ} \)

Calculate \( \angle A \) using the angle sum property of triangles: \( \angle A = 180^{\circ} - \angle B - \angle C \approx 180^{\circ} - 45.66^{\circ} - 38^{\circ} \approx 96.34^{\circ} \).

Step 5: Solve for Side a in the First Triangle

Apply the Law of Sines to find \( a \): \( \frac{a}{\sin A} = \frac{b}{\sin B} \), so \( \frac{a}{\sin 96.34^{\circ}} = \frac{45}{\sin 45.66^{\circ}} \). Solving gives \( a \approx 61.9 \).

Step 6: Evaluate the Second Triangle with \( \angle B \approx 134.34^{\circ} \)

If \( \angle B = 134.34^{\circ} \), then \( \angle A = 180^{\circ} - 134.34^{\circ} - 38^{\circ} \approx 7.66^{\circ} \).

Step 7: Solve for Side a in the Second Triangle

Using Law of Sines again: \( \frac{a}{\sin A} = \frac{b}{\sin B} \), so \( \frac{a}{\sin 7.66^{\circ}} = \frac{45}{\sin 134.34^{\circ}} \). Solve to find \( a \approx 6.01 \).

Step 8: Conclusion of Solutions

Two possible triangles satisfy the conditions: 1. \( \angle A \approx 96.34^{\circ}, \angle B \approx 45.66^{\circ}, a \approx 61.9 \).2. \( \angle A \approx 7.66^{\circ}, \angle B \approx 134.34^{\circ}, a \approx 6.01 \).

Key Concepts

Sine Function

Understanding the sine function is crucial when working with triangles and the Law of Sines. The sine function relates the angles of a triangle to the ratios of its sides. For a given angle \( \theta \) in a right triangle, the sine of \( \theta \) is defined as the ratio of the length of the opposite side to the hypotenuse in the function:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

This definition extends beyond right triangles when using the Law of Sines, where it allows the calculation of unknown angles or side lengths by relating all angles of the triangle to their opposite sides:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

The sine function is periodic, meaning it repeats its values in a regular cycle. This characteristic results in certain angles sharing the same sine value, such as \( \sin(45^\circ) = \sin(135^\circ) \). This is helpful when determining multiple solutions within triangle problems.

Triangle Angle Sum Property

A key property of triangles is that the sum of the internal angles always equals \(180^\circ\). Understanding this property is essential for solving triangles when given partial information such as two side lengths and an angle, as seen in our exercise:

• \( \angle A + \angle B + \angle C = 180^\circ \)

When you know two angles, the third can be calculated by subtracting their sum from \(180^\circ\). This property plays a crucial role in determining unknown angles in the solutions:

• e.g., \( \angle A = 180^\circ - \angle B - \angle C \)

In our exercise, when calculating \( \angle A \), we used this property after finding \( \angle B \) using the Law of Sines. This consistently balances the calculations required for verifying the feasibility of each possible triangle solution.

Solving Triangles

Solving triangles often involves identifying unknown sides or angles using given data. The Law of Sines and triangle properties are both essential tools in the solving process. In our scenario, we were given sides \( b \), \( c \), and angle \( \angle C \). Here is the general approach to solving such problems:

• Utilize the Law of Sines to find missing angles. Start with a given angle and opposing side information to solve for other angles.
• Check for possible solutions. Due to the nature of sine, sometimes there are two potential solutions (acute and obtuse angles) for one unknown value.
• Apply the triangle angle sum property to determine the remaining unknown angle.
• Use the Law of Sines again to find other unknown side lengths once all angles are known.

By systematically applying these steps, we uncovered two possible triangles that met the conditions:

• Triangle 1 with \( \angle A \approx 96.34^\circ \), \( \angle B \approx 45.66^\circ \), and side \( a \approx 61.9 \)
• Triangle 2 with \( \angle A \approx 7.66^\circ \), \( \angle B \approx 134.34^\circ \), and side \( a \approx 6.01 \)

Understanding this approach will enhance your ability to resolve complex triangle-based problems efficiently.