Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\beta=20^{\circ}, \quad \gamma=31^{\circ}, \quad b=210$$

Step-by-Step Solution

Answer

Angle α is 129°, side c is approximately 316.2, and side a is approximately 477.0.

Step 1: Find the Value of Angle α

Since the sum of the angles in a triangle is always 180 degrees, you can determine the unknown angle \( \alpha \) by using the equation:\[\alpha = 180^\circ - \beta - \gamma\]Substituting the known values:\[\alpha = 180^\circ - 20^\circ - 31^\circ = 129^\circ\]

Step 2: Use Law of Sines to Find Side c

The Law of Sines states that \( \frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} \).We need to find side \( c \), thus rewriting the equation:\[c = \frac{b \cdot \sin \gamma}{\sin \beta}\]Substituting the known values:\[c = \frac{210 \cdot \sin 31^\circ}{\sin 20^\circ}\]Calculate \( c \) using a calculator:\[c \approx \frac{210 \cdot 0.5150}{0.3420} \approx 316.2\]

Step 3: Use Law of Sines to Find Side a

Similarly, use the Law of Sines to find side \( a \):\[a = \frac{b \cdot \sin \alpha}{\sin \beta}\]Substitute the known values:\[a = \frac{210 \cdot \sin 129^\circ}{\sin 20^\circ}\]Calculate \( a \) using a calculator:\[a \approx \frac{210 \cdot 0.7771}{0.3420} \approx 477.0\]

Key Concepts

Law of Sines

One of the most powerful tools in trigonometry for solving triangles is the Law of Sines. This principle allows us to find unknown sides or angles of a triangle when we have some of the sides or angles already. The law states:\[\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}\]Here, \(a\), \(b\), and \(c\) are the sides opposite to angles \(\alpha\), \(\beta\), and \(\gamma\) respectively. This formula helps us compare the ratios of angles to their opposite sides, providing a way to solve for unknown values.When applying the Law of Sines, follow these guidelines:

• Identify the known angles and sides first.
• Choose the appropriate equation based on what you need to find.
• Simplify the formula to solve for the unknown side or angle.

In our specific problem, using the Law of Sines allowed us to find the unknown sides \(a\) and \(c\) effectively once we had the angle \(\alpha\). The key is to use the sine function of the known angles and relate them to the given side, \(b\). This systematic approach ensures minimal mistakes and accurate results, making trigonometry problems much more approachable.

Triangle Angle Sum

The triangle angle sum property is a fundamental concept in geometry and trigonometry. It states that the sum of the interior angles of any triangle is always 180 degrees. This property is crucial in determining unknown angles when two angles are given.To effectively use this property, remember:

• Add up the known angles.
• Subtract their sum from 180 degrees to find the missing angle.

In our problem, we know \(\beta = 20^\circ\) and \(\gamma = 31^\circ\). With the total being 180 degrees, we simply subtract the sum of these given angles from 180. This gives us the measure for angle \(\alpha\), which is 129 degrees.This simple calculation is often overlooked by students, but it is pivotal in trigonometry. Understanding the triangle angle sum property gives you a strong foundation to approach more complex problems and ensure you have all necessary information to use other trigonometric laws, like the Law of Sines.

Solving Triangles

Solving triangles involves finding all unknown sides and angles of a triangle, given some of its elements. It incorporates various trigonometric properties and laws. Here, using the triangle angle sum property and the Law of Sines, we can efficiently solve any triangle.The basic steps for solving triangles are:

• Identify and label all given parts of the triangle, such as sides and angles.
• Use the triangle angle sum property to find any missing angles.
• Apply the Law of Sines or other trigonometric laws to find unknown sides or angles.
• In cases where two sides and an angle are known, you can also use the Law of Cosines, not covered here.
• Double-check your calculations using a scientific calculator to ensure accuracy.

In the example problem, after determining angle \(\alpha\) using the triangle angle sum, we applied the Law of Sines to find sides \(c\) and \(a\). This process highlights how understanding each component of a triangle provides the groundwork to solve for the unknowns quickly and confidently.