Problem 18 - Precalculus

Question

The measures of two sides and an angle are given. Determine whether a triangle (or two) exist, and if so, solve the triangle(s). $$b=30, c=20, \beta=70^{\circ}$$

Step-by-Step Solution

Answer

No triangle exists as \( \sin(\gamma) > 1 \), which is impossible.

Step 1: Examine the Given Information

You are provided with the measures of two sides and the angle between them. Here, the sides are given as \( b = 30 \) and \( c = 20 \) with the angle \( \beta = 70^{\circ} \). This scenario suggests the use of the Law of Sines to determine the number of possible triangles.

Step 2: Determine Possible Triangles Using the Law of Sines

First, apply the Law of Sines to find \( \sin(\gamma) \), where \( \gamma \) is the angle opposite side \( c \): \[ \frac{\sin(\gamma)}{b} = \frac{\sin(\beta)}{c} \]Substitute the given values:\[ \frac{\sin(\gamma)}{30} = \frac{\sin(70^{\circ})}{20} \]Calculating \( \sin(70^{\circ}) \approx 0.9397 \):\[ \sin(\gamma) = \frac{30 \times 0.9397}{20} \approx 1.4096 \]Since \( \sin(\gamma) > 1 \), which is not possible for any real angle \( \gamma \), no triangle can exist with these dimensions.

Key Concepts

Trigonometry in Triangles

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It provides tools to solve problems involving triangles, especially right-angled triangles, using ratios like sine, cosine, and tangent. These trigonometric ratios are essential in establishing the formulas used to solve triangles. For any triangle, these ratios help in understanding how the lengths of the sides relate to the angles.
For instance, when we're given particular measurements in a triangle, trigonometry allows us to predict the missing sides or angles.
Formulas, such as the Law of Sines and the Law of Cosines, derive directly from these trigonometric ratios. These are particularly helpful in evaluating triangles that are not right-angled, by using known quantities to find unknown values, making it possible to "solve" the triangle completely.
Trigonometry is extensively applied in various fields such as engineering, physics, and astronomy, highlighting its practical significance.

Determining Triangle Existence

Not all combinations of side lengths and angles will form a valid triangle. The principle of triangle existence requires certain conditions to be met. The measures of the angles and sides must make logical sense to form a true triangle.
In many cases, it's helpful to use inequalities, like the triangle inequality theorem, to check if a triangle can exist. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

• This criterion helps to eliminate impossible scenarios right away.
• However, when angles are involved, as in this exercise, the Law of Sines can indicate whether a solution is plausible.

In the given scenario where the calculated sine of an angle came out as greater than one, it was clear cut that no such triangle could exist.
This is because the sine of any angle cannot exceed one, making the scenario inconsistent with the rules of trigonometry.

Understanding the Law of Sines

The Law of Sines is a crucial tool in trigonometry for solving triangles, especially when dealing with non-right triangles. This law relates the sides of a triangle to the sines of its angles.
It states:

• \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \)

This formula allows us to solve for unknown angles or sides when we have partial information about the triangle. For instance, if two sides and a non-included angle are known, the Law of Sines can determine the potential existence of a solution.
However, sometimes it reveals that no triangle can form with the given data—as was the case in the provided exercise.

• The calculated value of sine for one of the angles resulted in a number outside its permissible range.
• This law not only helps in finding missing sides and angles but also acts as a check to ensure the triangle's existence aligns with trigonometric principles.

By understanding the Law of Sines, we respect the boundaries set by trigonometry, ensuring our solutions are both accurate and valid.