Problem 4 - Precalculus with Calculus Previews

Question

Use the Law of Cosines to solve the triangle. $$ \gamma=31.5^{\circ}, a=4, b=8 $$

Step-by-Step Solution

Answer

The side \( c \approx 5.013 \), \( \alpha \approx 24.63^\circ \), and \( \beta \approx 123.87^\circ \).

Step 1: Understand the Problem

We are given a triangle with an angle \( \gamma = 31.5^\circ \) and sides \( a = 4 \) and \( b = 8 \). We need to find the remaining side \( c \) and the other two angles \( \alpha \) and \( \beta \).

Step 2: Use the Law of Cosines

The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma) \]. Substituting in the given values: \[ c^2 = 4^2 + 8^2 - 2 \cdot 4 \cdot 8 \cdot \cos(31.5^\circ) \].

Step 3: Calculate \( c \)

Compute \( c \) using the formula: \[ c^2 = 16 + 64 - 64 \cdot \cos(31.5^\circ) \]. Simplifying, we find \( c^2 = 80 - 64 \cdot 0.8572 \). Thus, \( c^2 = 80 - 54.8608 = 25.1392 \), so \( c = \sqrt{25.1392} \approx 5.013 \).

Step 4: Find \( \alpha \) using the Law of Sines

Use the Law of Sines: \[ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \]. Solving for \( \alpha \): \[ \sin(\alpha) = \frac{a \cdot \sin(\gamma)}{c} = \frac{4 \cdot \sin(31.5^\circ)}{5.013} \]. Thus, \( \sin(\alpha) \approx \frac{4 \cdot 0.5225}{5.013} \approx 0.417 \), so \( \alpha \approx \sin^{-1}(0.417) \approx 24.63^\circ \).

Step 5: Calculate \( \beta \)

Since the sum of the angles in a triangle is always \( 180^\circ \), we have \[ \beta = 180^\circ - \gamma - \alpha = 180^\circ - 31.5^\circ - 24.63^\circ \]. Thus, \( \beta \approx 123.87^\circ \).

Step 6: Final Solution Check

Verify if the calculated values satisfy all laws, especially checking if \( a^2 = b^2 + c^2 - 2bc \cdot \cos(\alpha) \) and similar for \( \beta \). After confirmation, the final values appear accurate.

Key Concepts

Triangle Solution

Solving a triangle involves determining all of its components, such as side lengths and angles, using given information. In this exercise, we were initially provided with one angle (\( \gamma = 31.5^\circ \)) and two sides (\( a = 4 \) and \( b = 8 \)).
This represents a typical problem in trigonometry where you need to find the remaining side, \( c \), and the angles \( \alpha \) and \( \beta \), using known values. We apply trigonometric laws such as the Law of Cosines and the Law of Sines to solve for unknown parts of the triangle.
Here's a quick rundown of what was done:

• Use the Law of Cosines to find the unknown side \( c \).
• Apply the Law of Sines to determine the angle \( \alpha \).
• Calculate the final angle \( \beta \) using the triangle sum property, which states that the sum of all angles in a triangle is \( 180^\circ \).

Through these steps, you can fully solve any triangle when given sufficient initial data including at least one side and one angle.

Trigonometry

Trigonometry is a branch of mathematics that explores relationships involving lengths and angles of triangles. It's a powerful tool for solving problems in various fields such as physics, engineering, and even computer graphics. In triangle solutions, trigonometry allows us to calculate unknown sides and angles if certain elements are given.
Trigonometry uses functions like sine, cosine, and tangent to relate the sides and angles of a triangle. These functions are essential for understanding how to apply different trigonometric laws, like the Law of Cosines and the Law of Sines, for determining missing components in a triangle.
When working with angles, you often use degrees or radians as measurement units. It's important to consistently use one unit to avoid calculation errors. In our exercise, we are working with degrees, which is a common practice in solving triangles where angles are mentioned explicitly. Understanding these fundamentals is key to mastering trigonometry.

Law of Sines

The Law of Sines establishes a relationship between the sides of a triangle and the sines of its angles, providing an efficient method to solve for unknown angles or sides when certain criteria are met. It is defined as:\[\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\]This law is especially useful when you know:

• Two angles and one side (AAS or ASA) to find the remaining side.
• Two sides and one non-enclosed angle (SSA) to find the remaining angle.

For our triangle problem, once we found side \( c \) with the Law of Cosines, we used the Law of Sines to easily calculate the angle \( \alpha \). By rearranging the formula to solve for \( \alpha \), we used the sine function to determine its measure in degrees.
Understanding how to manipulate and apply the Law of Sines is invaluable for any student aiming to master triangle solutions using trigonometry.