Problem 14 - Precalculus Mathematics for Calculus

Question

Solve triangle \(A B C\). \(b=60, \quad c=30, \quad \angle A=70^{\circ}\)

Step-by-Step Solution

Answer

The triangle cannot exist under conventional geometric conditions.

Step 1: Determine Angle B Using Law of Sines

We will use the Law of Sines, which is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). First, we need to find angle \( B \). Since we already know angle \( A \) and sides \( b \) and \( c \), we can use \( \frac{b}{\sin B} = \frac{c}{\sin A} \) to find \( \sin B \). This gives us: \( \sin B = \frac{b \cdot \sin A}{c} = \frac{60 \cdot \sin 70^{\circ}}{30} \). Calculating this gives \( \sin B \approx 1.8790 \), which is greater than 1. Thus, we cannot directly calculate \( B \) as the assumed configuration leads to a logical inconsistency. We might need to reconsider or use another approach, like determining angle \( C \) first.

Step 2: Determine Angle C

Since we discovered step 1 results in an inconsistency, we need to determine angle \( C \) first. The angle sum of a triangle is \( 180^{\circ} \). Therefore, \( \angle C = 180^{\circ} - \angle A = 180^{\circ} - 70^{\circ} = 110^{\circ} \).

Step 3: Determine Angle B and Check Assumptions

We now have \( \angle C \). By checking assumptions, the angle \( B \) can be calculated from the angle sum of the triangle: \( \angle B = 180^{\circ} - \angle A - \angle C = 180^{\circ} - 70^{\circ} - 110^{\circ} = 0^{\circ} \). Spotting a logical inconsistency in assumptions, angle \( B \) cannot be zero in a valid triangle configuration.

Step 4: Correctly Determine The Configuration

Since the direct calculations provided impossible results, it should be concluded the triangle does not exist as prescribed if conventional conditions are considered. The triangle configuration presupposed does not mathematically formulate a valid triangle.

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry used to solve triangles, especially those that are not right-angled. This law states that the ratio of a side length to the sine of its opposite angle is constant for all three sides in a triangle.
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] It enables us to find unknown angles or sides when dealing with triangle problems. In the case of triangle \( ABC \), if we know two sides and an angle, we can use the Law of Sines to find missing information.

• Begin with known values.
• Solve for \( \sin B \) or \( \sin C \) using known side and angle values.

However, as noted, calculations might result in logical inconsistencies, such as obtaining a sine value greater than 1, which is mathematically impossible. This indicates errors in assumptions or the triangle's existence under given conditions.

Triangle Properties

Understanding triangle properties is essential in the study of trigonometry. A triangle is generally defined by its three sides and three angles. Here are some key properties:

• The sum of the angles in any triangle is always \(180^{\circ}\).
• In a valid triangle, each angle must be less than \(180^{\circ}\).
• A triangle is non-existent if any angle results in a value of \(0^{\circ}\) or \(180^{\circ}\).

When solving triangles, it's crucial to ensure the properties are fulfilled. For triangle \(ABC\), the inconsistency arose as angle \( B \) was calculated to be \(0^{\circ}\), challenging the triangle's validity. This highlights the importance of careful assessment of triangle constraints.

Angle Sum Property

The angle sum property of a triangle is a fundamental axiom asserting that the sum of all internal angles in a triangle is \(180^{\circ}\). This property is pivotal when determining unknown angles.
In the exercise, we used this property to try to find missing angle \( C \) first, which was \(110^{\circ}\), by subtracting the known angle \( A \) from \(180^{\circ}\). Once \( C \) was calculated, we attempted to find \( B \), resulting in a contradiction.
The angle sum property ensures that when known angles are subtracted from \(180^{\circ}\), the remaining must be plausible. A triangle cannot exist if one angle results in \(0^{\circ}\) or exceeds logical bounds, emphasizing the necessity for valid angle assumptions when solving triangles.