Problem 51 - Precalculus

Question

Solve the triangle \(\alpha=120^{\circ}, a=7,\) and \(b=9\) Solution: Use the Law of sines to find \(\beta .\) \(\frac{\sin \alpha}{a}=\frac{\sin \beta}{b}\) Let \(\alpha=120^{\circ}, a=7\) and \(b=9\). \(\frac{\sin 120^{\circ}}{7}=\frac{\sin \beta}{9}\) Solve for \(\sin \beta\). \(\sin \beta=1.113\) Solve for \(\beta\). \(\beta=42^{\circ}\) Sum the angle measures to \(180^{\circ}\). \(120^{\circ}+42^{\circ}+\gamma=180^{\circ}\) Solve for \(\gamma\). \(\gamma=18^{\circ}\) Use the Law of sines to find \(c\). \(\frac{\sin \alpha}{a}=\frac{\sin \gamma}{c}\) Let \(\alpha=120^{\circ}, a=7\) and \(\gamma=18^{\circ}\). \(\frac{\sin 120^{\circ}}{7}=\frac{\sin 18^{\circ}}{c}\) Solve for \(c\). \(c=2.5\) \(\alpha=120^{\circ}, \beta=42^{\circ}, \gamma=18^{\circ}, a=7, b=9,\) and \(c=2.5\) This is incorrect. The longest side is not opposite the longest angle. There is no triangle that makes the original measurements work. What mistake was made?

Step-by-Step Solution

Answer

The triangle is invalid as \(\sin \beta > 1\) shows no angle corresponds. Thus, the triangle doesn't exist with given measures.

Step 1: Analyze Given Information

We have a triangle with known values: \(\alpha = 120^{\circ}\), \(a = 7\), and \(b = 9\). We use the Law of Sines to proceed.

Step 2: Apply the Law of Sines

Use the Law of Sines formula: \(\frac{\sin \alpha}{a} = \frac{\sin \beta}{b}\). Substitute the given values to get: \(\frac{\sin 120^{\circ}}{7} = \frac{\sin \beta}{9}\).

Step 3: Calculate \(\sin \beta\)

First, find \(\sin 120^{\circ} = \sqrt{3}/2 \approx 0.866\). Calculate \(\sin \beta\): \(\sin \beta = 9 \times \frac{0.866}{7} \approx 1.113\).

Step 4: Recognize Error in Calculation

The value \(\sin \beta = 1.113\) is invalid because the sine of an angle cannot exceed 1. This indicates an error due to incorrect assumptions about the triangle.

Step 5: Reassess Triangle Validity

The calculation mistake suggests that the triangle's properties do not align for \(\alpha = 120^{\circ}\), \(a = 7\), and \(b = 9\). Verify whether such a triangle can exist.

Step 6: Evaluate Triangle Configuration

A triangle with \(\alpha = 120^{\circ}\) suggests that the side opposite \(\alpha\) should be the longest (side \(a\)) but given \(b = 9\) (longer than \(a\)), the configuration is impossible under these rules.

Key Concepts

Triangle Existence

When dealing with triangle problems, it's crucial to understand the concept of triangle existence. In geometry, a triangle is only possible if it satisfies certain conditions. One fundamental rule is the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. This means that for a triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

If any of these conditions is violated, then the three sides cannot form a triangle. In the context of the given exercise, the sides \(a=7\) and \(b=9\) with an opposite angle of \(\alpha = 120^{\circ}\) suggest that side \(a\) should be the longest, opposite the largest angle. However, side \(b\) being longer contradicts this, signaling the impossibility of this triangle. Understanding and applying these conditions ensures the existence of a valid triangle before proceeding with calculations.

Angle Calculation

Determining angles in a triangle is a vital part of solving triangles. The Law of Sines is often used, which relates the sides of a triangle to their opposite angles. The Law of Sines formula is expressed as: \[\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}\] When given two sides and an angle (SSA), additional care must be taken since it can sometimes lead to ambiguous cases or no solution at all. We start by calculating the sine of the given angle. For example, \(\sin 120^{\circ} = \sqrt{3}/2\). When substituting these into the formula, errors can occur, as seen when \(\sin \beta\) was calculated as 1.113, which is impossible since the sine of any angle cannot exceed 1. Knowing when such errors occur is critical in detecting whether there's a problem with the inputs being invalid for a triangle. When everything checks out, the rest of the angles in the triangle can be found by ensuring their sum is \(180^{\circ}\). Hence, angle calculation needs a combination of algebra and understanding of trigonometric principles.

Sine Function Validity

The sine function represents the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. Its range, however, is constrained between \(-1\) and \(1\) inclusive. This critical range limits what values the sine of an angle can actually equal. Any attempt to calculate a sine value outside of this range immediately indicates a mistake, usually suggesting incorrect assumptions or invalid triangle configurations. Common mistakes include assuming an angle can have a sine greater than 1, as seen when \(\sin \beta = 1.113\) was proposed. Such calculations are patently false since they fall outside the sine function's valid range. Recognizing such errors quickly can prevent further false solutions from being explored. The function's limitations are essential to identifying impossible or erroneous conditions in triangle problems. Always verify trigonometric results against these known principles to ensure they are logically sound.