Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Solve \(\triangle A B C\). $$\alpha=32.32^{\circ}, \quad c=574.3, \quad a=263.6$$

Step-by-Step Solution

Answer

\( \beta = 77.32^{\circ}, \gamma = 70.36^{\circ}, \text{and } b \approx 508.0 \).

Step 1: Identify the known values

We have been given \( \alpha = 32.32^{\circ} \), \( c = 574.3 \), and \( a = 263.6 \) in triangle \( \triangle ABC \). We aim to find \( \beta \), \( \gamma \), \( b \), and \( A \).

Step 2: Apply the law of sines to find \( \beta \)

Using the Law of Sines, \( \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \). We can solve for \( \sin(\beta) \) using \( \frac{a}{\sin(32.32^{\circ})} = \frac{c}{\sin(\beta)} \) to find \( \beta \).

Step 3: Solve for \( \sin(\beta) \)

Substitute our known values: \( \frac{263.6}{\sin(32.32^{\circ})} = \frac{574.3}{\sin(\beta)} \).Using trigonometric calculations, solve for \( \sin(\beta) \):\[ \sin(\beta) \approx \frac{574.3 \cdot \sin(32.32^{\circ})}{263.6} \].

Step 4: Determine \( \beta \)

Calculate \( \beta \) using the inverse sine function: \( \beta = \sin^{-1}(\sin(\beta)) \). Using a calculator, we find \( \beta \approx 77.32^{\circ} \).

Step 5: Find \( \gamma \) using angle sum property

Use the angle sum property of triangles: \( \gamma = 180^{\circ} - \alpha - \beta = 180^{\circ} - 32.32^{\circ} - 77.32^{\circ} \). Calculating gives \( \gamma \approx 70.36^{\circ} \).

Step 6: Calculate \( b \) using Law of Sines

Use the Law of Sines again: \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \). Solve for \( b \): \[ b = \frac{263.6 \cdot \sin(77.32^{\circ})}{\sin(32.32^{\circ})} \] to find \( b \approx 508.0 \).

Step 7: Verify the results

Sum the angles we found: \( \alpha = 32.32^{\circ} \), \( \beta = 77.32^{\circ} \), and \( \gamma = 70.36^{\circ} \). Check if they add to \( 180^{\circ} \): \( 32.32^{\circ} + 77.32^{\circ} + 70.36^{\circ} = 180^{\circ} \). This confirms our solution is consistent.

Key Concepts

Angle Sum Property of Triangles

The angle sum property of triangles is a fundamental rule in geometry that states the sum of the interior angles of a triangle is always 180 degrees. This property is extremely helpful when solving triangles, especially when some angles are unknown.

For instance, if you know two angles in a triangle, the third can be easily calculated by subtracting the sum of the two known angles from 180 degrees. This was used effectively in our exercise, where we calculated angle \( \gamma \) using the formula \( \gamma = 180^{\circ} - \alpha - \beta \).

Understanding this property is crucial as it serves as a checkpoint for ensuring any triangle calculations are accurate. By confirming that the sum of your calculated angles equals 180 degrees, you verify that there has been no mistake in your prior calculations.

Inverse Sine Function

The inverse sine function, often written as \( \sin^{-1} \) or arcsin, is used to determine an angle when the value of its sine is known. This is especially pertinent in trigonometry when solving triangles, as it allows us to find an unknown angle given a known ratio.

In our exercise, once we found the value for \( \sin(\beta) \), we used the inverse sine function to calculate \( \beta \). This is done by \( \beta = \sin^{-1}(\sin(\beta)) \). It's essential to use a calculator for this part, as the function requires calculated precision, often yielding results in degrees or radians.

The inverse functions are key components in trigonometry, allowing angle determination from known trigonometric values, such as sine, cosine, or tangent.

Trigonometry Calculations

Trigonometry calculations often involve functions that relate the angles and sides of triangles. In solving triangles, the Law of Sines is frequently used, which establishes the relationship between the sides and angles of a general, non-right triangle.

The Law of Sines states that for a triangle with sides \( a \), \( b \), and \( c \) opposite angles \( \alpha \), \( \beta \), and \( \gamma \) respectively, \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \).

In our problem, this was used to solve for both \( \beta \) and \( b \). By rearranging the terms, we solve for the unknowns systematically.

These calculations underpin not just solving triangles but also aid in various applications, including physics and engineering, where similar triangles appear.