Problem 5 - Technical Mathematics with Calculus

Question

Solve triangle \(A B C\). $$A=44.47^{\circ}, \quad C=63.88^{\circ}, \quad c=1.065$$

Step-by-Step Solution

Answer

Angle B is 71.65 degrees. Side a is approximately 0.745 units, and side b is approximately 0.951 units.

Step 1: Calculate angle B

The sum of angles in a triangle is always 180 degrees. Since angles A and C are given, angle B can be calculated by subtracting the sum of angles A and C from 180 degrees: \(B = 180^{\text{°}} - A - C\).

Step 2: Apply the Law of Sines to find side a

Using the Law of Sines, \(\frac{a}{\sin(A)} = \frac{c}{\sin(C)}\), we can solve for side a by rearranging the equation: \(a = \frac{c \cdot \sin(A)}{\sin(C)}\).

Step 3: Apply the Law of Sines to find side b

Similarly, using the Law of Sines, \(\frac{b}{\sin(B)} = \frac{c}{\sin(C)}\), we can find side b by rearranging the equation: \(b = \frac{c \cdot \sin(B)}{\sin(C)}\).

Key Concepts

Law of Sines

The Law of Sines is an essential tool in trigonometry, particularly when dealing with non-right triangles. It provides a relationship between the lengths of sides and the sines of the opposite angles in a triangle.

It is often expressed as \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\), where \(a, b,\) and \(c\) are the lengths of the sides, and \(A, B,\) and \(C\) are the opposite angles respectively. This formula allows us to find unknown lengths or angles when we have enough information from the rest of the triangle.

For instance, if two angles and one side are known, as in the example exercise, we can easily find the remaining sides by isolating the unknown side in the equation and multiplying both sides by the sine of its opposite angle, thus solving for the unknown side. This powerful application of the Law of Sines makes it a cornerstone in triangle solving strategies.

Triangle Angle Sum

The triangle angle sum is a fundamental theorem in geometry that states the sum of the three interior angles in any triangle is always \(180^\circ\). This simple yet profound rule is crucial for problem-solving in geometry, especially when one of the goals is to find an unknown angle.

How to Utilize the Triangle Angle Sum

In practice, to find a missing angle in a triangle, you simply subtract the known angles from \(180^\circ\). This is the first pivotal step in our example exercise where we determined the third angle by using the given angles. Ensuring students understand this concept is paramount, as it is the foundation for solving various types of triangle-related problems that they will encounter both inside and outside the classroom.

Sine Function

Understanding the sine function is integral to mastering trigonometry and its applications. The sine function relates an angle in a right triangle to the ratio of the length of the side opposite that angle to the length of the longest side, or the hypotenuse.

In the context of triangles, specifically for solving purposes, the sine function helps in relating the angles to their opposite sides. It becomes even more relevant when the Law of Sines is involved, as it is a direct representation of the ratio that the law compares.

Real-World Applications

The sine function is not just theoretical; it is widely used in fields such as physics, engineering, and geography. For example, in physics, it is used to calculate the components of force or velocity. In creating our solutions, it is vital to comprehend how the sine of an angle is determined and to be able to calculate it accurately when given a degree or radian measure. Ensuring students grasp this will open doors to solving complex problems and understanding the sine's practical uses beyond the classroom.