Problem 8 - Precalculus with Calculus Previews

Question

Use the Law of Sines to solve the triangle. $$ \alpha=30^{\circ}, \gamma=75^{\circ}, a=6 $$

Step-by-Step Solution

Answer

The triangle has angles \(30^\circ, 75^\circ, 75^\circ\), with sides \(a = 6, b \approx 11.59, c \approx 11.59\).

Step 1: Calculate the Missing Angle

In any triangle, the sum of angles is always 180 degrees. To find the missing angle \( \beta \), subtract the given angles from 180 degrees: \( \beta = 180^\circ - \alpha - \gamma \). Substituting \( \alpha = 30^\circ \) and \( \gamma = 75^\circ \), we get \( \beta = 180^\circ - 30^\circ - 75^\circ = 75^\circ \).

Step 2: Use the Law of Sines to Find Side b

The Law of Sines states that \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \). We need to find side \( b \), so use the equation \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \). Thus, \( b = \frac{a \cdot \sin \beta}{\sin \alpha} \). Substitute \( a = 6 \), \( \alpha = 30^\circ \), and \( \beta = 75^\circ \) to find \( b = \frac{6 \cdot \sin 75^\circ}{\sin 30^\circ} \approx \frac{6 \cdot 0.9659}{0.5} \approx 11.59 \).

Step 3: Use the Law of Sines to Find Side c

Again using the Law of Sines \( \frac{a}{\sin \alpha} = \frac{c}{\sin \gamma} \), solve for \( c \): \( c = \frac{a \cdot \sin \gamma}{\sin \alpha} \). Substitute \( a = 6 \), \( \alpha = 30^\circ \), and \( \gamma = 75^\circ \) to find \( c = \frac{6 \cdot \sin 75^\circ}{\sin 30^\circ} \approx 11.59 \).

Key Concepts

Triangle Solution

When solving a triangle, our aim is to determine all its missing angles and sides based on given information. In our case, we have two angles and one side provided. These triangles are classified as an ASA (Angle-Side-Angle) type. By using known angles, we can discover the third, and with one known side, we can apply trigonometric principles to find the others. Always start by calculating any missing angles, followed by determining the unknown sides. This sequence simplifies workflows, allowing one to correctly apply the Law of Sines to complete the triangle information.

• First, calculate the missing angle.
• Then, use the Law of Sines for other sides.

Trigonometry

Trigonometry is the branch of mathematics dealing with triangles, especially right-angled ones. The Law of Sines is a crucial formula here and applies to any type of triangle. It relates angles of a triangle to the lengths of its sides. This law is expressed as: \[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]This relation is incredibly useful because it allows us, given some known values, to unravel the rest of the triangle. Besides triangle solutions, trigonometry has broader applications:

• Calculating distances in astronomy.
• Computer graphics.
• Navigation and geography.

Understanding trigonometry, thus, opens various doors to solving real-world problems.

Angle Calculation

Calculating angles in a triangle often begins with the fundamental rule that a triangle’s internal angles always sum up to 180 degrees. Knowing this, we can calculate any missing angle by subtracting the sum of the other two from 180:\[ \beta = 180^\circ - \alpha - \gamma \]Once you identify your situation: perhaps an ASA triangle, calculate the unknown angle first before proceeding. This approach ensures accuracy in further calculations, such as finding missing sides. In our solution, we discovered angle \(\beta\) to be 75 degrees using:

• \(\alpha = 30^\circ\).
• \(\gamma = 75^\circ\).
• Then \(\beta = 75^\circ\).

Such steps, though elementary, build the foundation of more complex trigonometric solutions later.

Side Calculation

Side calculation leverages the Law of Sines, where previously determined angles come into play. Knowing one side and all angles, use:\[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]To calculate side \(b\), rearrange to:\[ b = \frac{a \cdot \sin \beta}{\sin \alpha} \]Substitute your known values \(a = 6\), \(\alpha = 30^\circ\), and \(\beta = 75^\circ\):

• Calculate \(\sin 30^\circ\) and \(\sin 75^\circ\).
• \(b \approx \frac{6 \cdot 0.9659}{0.5} \approx 11.59\).
• Repeat the process for other sides.

Achieving side calculations accurately hinges on precise trigonometric values, ensuring that solutions hold true in practical scenarios.