Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Explain why no triangle \(A B C\) exists having \(A=103^{\circ}, a=12, b=13\)

Step-by-Step Solution

Answer

This triangle cannot exist because \(\sin B\) exceeds 1, violating the range of the sine function.

Step 1: Establish the triangle's angle sum property

Start by recalling that the sum of angles in any triangle is always 180 degrees. Since angle \(A\) is 103 degrees, the sum of angles \(B\) and \(C\) must be 77 degrees.

Step 2: Apply the Law of Sines

Using the Law of Sines, \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Plug in the values for \(a\), \(b\), and \(A\): \( \frac{12}{\sin 103^{\circ}} = \frac{13}{\sin B} \).

Step 3: Solve the equation for \(\sin B\)

Calculate \(\sin 103^{\circ}\) and substitute it back to find \(\sin B\): \( \sin B = \frac{13 \times \sin 103^{\circ}}{12} \). Since \(103^{\circ}\) is an obtuse angle, \(\sin 103^{\circ}\) is positive and less than 1.

Step 4: Check feasibility for \(\sin B\) within possible range

Calculate \(\sin B\): if \( \sin B > 1 \), then such a triangle cannot exist, due to the range limit of the sine function, which is [-1,1].

Step 5: Conclude non-existence based on \(\sin B\)

Evaluating the value calculated in Step 3 reveals \(\sin B\) exceeds 1, indicating \(B\) is impossible to achieve, confirming such triangle \(ABC\) can't exist with given parameters.

Key Concepts

Law of Sines

The Law of Sines is a key concept when working with triangles, particularly in situations where some angles and sides are known, and others are not. It states that in any triangle, the ratio of the length of a side to the sine of its corresponding angle is constant across all three sides and angles of the triangle. Mathematically, this is expressed as:

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

This formula is particularly useful when you know:

• Two angles and one side (AAS or ASA) and need to find another side.
• Two sides and a non-enclosed angle (SSA) and need to find an angle or the third side.

For example, given the problem's context:

• We know angle \(A = 103^{\circ}\) and sides \(a = 12\), \(b = 13\).
• Using the Law of Sines, \( \frac{12}{\sin 103^{\circ}} = \frac{13}{\sin B} \).

Solving this helps to understand if a triangle configuration is possible based on trigonometric limits. If the calculated sine of an angle is greater than 1, such a triangle isn't possible.

Triangle Inequality

One critical property of triangles is the Triangle Inequality Theorem. This theorem asserts that for any triangle, the sum of any two sides must be greater than the third side. The implications are:

• For sides \(a\), \(b\), and \(c\) in a triangle, must always have: \( a + b > c \), \(a + c > b \), \(b + c > a \).

The Triangle Inequality is fundamental because it ensures that line segments can indeed form the sides of a triangle. For the triangle ABC specified, knowing this property can provide initial checks even before calculations:

• With \(a = 12\) and \(b = 13\), you would anticipate another side \(c\) that adheres to these constraints to form a triangle.

If any inequalities are not satisfied, such a triangle can't exist.

Angle Sum Property

A fundamental principle of triangle geometry is the Angle Sum Property, which states that the sum of the internal angles of a triangle is always 180 degrees. This helps in finding out unknown angles when the other angles are known.

• For example, in any triangle ABC, \(A + B + C = 180^{\circ}\).

In the original exercise, angle \(A\) is given as \(103^{\circ}\). Thus, \(B + C\) must sum up to \(77^{\circ}\) to satisfy the angle sum property:

• \(B + C = 77^{\circ} \).

This property is pivotal when paired with other geometric or trigonometric rules to solve for unknowns, commonly while applying the Law of Sines or Cosines.