Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\alpha=27^{\circ} 30^{\prime}, \quad c=52.8, \quad a=28.1$$

Step-by-Step Solution

Answer

Solve using Law of Sines; find \(\gamma\) and \(\beta\), then \(b\).

Step 1: Convert Angle to Decimal Form

First, convert the given angle \(\alpha = 27^{\circ} 30^{\prime}\) into a decimal. Since \(30\) minutes is \(0.5\) degrees, this becomes \(27.5^{\circ}\).

Step 2: Use Law of Sines

Using the Law of Sines, which states \(\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\). Here, we know \(a = 28.1\), \(c = 52.8\), and \(\alpha = 27.5^{\circ}\).

Step 3: Find \(\gamma\) Using the Law of Sines

First, solve for \(\sin(\gamma)\): \[\sin(\gamma) = \frac{c \cdot \sin(\alpha)}{a} = \frac{52.8 \cdot \sin(27.5^{\circ})}{28.1}.\] Calculate \(\sin(27.5^{\circ})\) using a calculator, then solve for \(\gamma\).

Step 4: Calculate \(\gamma\)

Calculate \(\gamma\) with \(\sin(\gamma)\) found in the previous step. Use a calculator to find \(\gamma = \sin^{-1}(\text{previous result})\).

Step 5: Determine \(\beta\)

Since we now know \(\alpha\) and \(\gamma\), use the angle sum property of triangles, \(\alpha + \beta + \gamma = 180^{\circ}\), to solve for \(\beta\). \(\beta = 180^{\circ} - \alpha - \gamma\).

Step 6: Find \(b\) Using the Law of Sines

With \(\beta\) known, use the Law of Sines again: \(\frac{b}{\sin(\beta)} = \frac{a}{\sin(\alpha)}\) to solve for \(b\). Calculate \(b = \frac{a \cdot \sin(\beta)}{\sin(\alpha)}\) to find the value of \(b\).

Key Concepts

Law of Sines Applications

The Law of Sines is a fundamental principle used to solve triangles, especially non-right triangles where direct trigonometric functions do not suffice. When a triangle is not right-angled, knowing two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) is necessary for applying the Law of Sines. This law allows us to relate the angles and sides of a triangle. It is expressed as:

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

Using this equation, you can solve for unknown sides or angles of a triangle when a sufficient amount of measurements are known. In the exercise, we used the Law of Sines to find the missing angle \( \gamma \) and then used it again to find the missing side \( b \). Overall, the Law of Sines is versatile for handling any case where a non-right triangle is involved, making it a handy tool in trigonometry problems.
Another important aspect of this formula is its ability to be rearranged based on the known quantities, enabling the solution of complex geometric problems by serial calculations.

Angle Conversion

Converting angle measurements is crucial in geometry, and specifically, it becomes necessary when solving triangle problems. In our exercise, the angle given was in degrees and minutes, \(27^{\circ} 30^{\prime}\). To convert it to a decimal format, which is often easier for calculations, we must recognize that one degree (ational{^\circ} 0) is equivalent to 60 minutes (ational {^\prime} ), akin to how hours convert to minutes.

• Thus, \( 30^{\prime} = 0.5^{\circ} \) (because \( \frac{30}{60} = 0.5 \)). The angle converts to \( 27.5^{\circ} \).

This step simplifies plugging values into trigonometric functions, easing the calculations within the context of solving triangles. Accuracy in this conversion is critical, as small errors can lead to significant discrepancies in results.

Triangle Properties

Triangles have fundamental properties that determine their nature and categorize their types. Among the properties, the sum of interior angles always being \(180^\circ\) is key. This property was used in our problem to find the third angle once two others were known.

• We can express this as \( \alpha + \beta + \gamma = 180^\circ \).

Another property to remember is that a larger angle is opposite a longer side, which helps predict the triangle's shape. Triangles are also classified based on their angles (acute, obtuse, right) and sides (equilateral, isosceles, scalene) which help in choosing the correct trigonometric principle for calculations. Understanding these properties ensures a structured approach in solving and understanding any question involving triangles. Always double-check these properties when solving complex problems to guide you to the right solution.