Solving Triangle Side Length And Complex Number Cartesian Form

This section delves into the application of the Law of Cosines to calculate the unknown side length of a triangle when two sides and the included angle are known. This is a fundamental concept in trigonometry and has widespread applications in various fields such as surveying, navigation, and engineering.

Understanding the Law of Cosines:

The Law of Cosines is a generalization of the Pythagorean theorem, relating the sides and angles of any triangle. It states that for a triangle with sides of lengths a, b, and c, and an angle γ opposite side c, the following equation holds:

c² = a² + b² - 2ab cos(γ)

Where:

• a, b, and c are the lengths of the sides of the triangle.
• γ is the angle opposite side c.

This law is particularly useful when dealing with non-right triangles, where the basic trigonometric ratios (sine, cosine, tangent) cannot be directly applied. In essence, the Law of Cosines bridges the gap between side lengths and angles in any triangle, making it a versatile tool for solving geometric problems.

Applying the Law of Cosines to the Triangular Plot Problem:

In our specific problem, we are given a triangular plot with two sides of lengths 70 meters and 50 meters, and the angle between them is 60 degrees. Our objective is to find the length of the third side. Let's denote the given sides as a = 70 meters and b = 50 meters, and the included angle as γ = 60 degrees. We want to find the length of the third side, which we'll call c.

Using the Law of Cosines formula, we have:

c² = 70² + 50² - 2 * 70 * 50 * cos(60°)

To proceed, we need to evaluate cos(60°). Recall that cos(60°) = 1/2. Substituting this value into the equation, we get:

c² = 4900 + 2500 - 2 * 70 * 50 * (1/2) c² = 7400 - 3500 c² = 3900

To find c, we take the square root of both sides:

c = √3900 c ≈ 62.45 meters

Therefore, the length of the third side of the triangular plot is approximately 62.45 meters.

Significance and Applications:

The Law of Cosines is a powerful tool in solving various geometric problems, particularly those involving triangles. Its applications extend beyond academic exercises and into real-world scenarios. For instance, surveyors use the Law of Cosines to determine distances and angles in irregular terrains. Navigators rely on it to calculate distances and bearings between locations. Engineers utilize it in structural design and analysis.

The ability to determine unknown side lengths and angles in triangles is crucial in many practical situations, making the Law of Cosines an indispensable tool for professionals in diverse fields. By understanding and applying this law, we can effectively solve a wide range of problems involving triangular shapes.

This section focuses on expressing a complex number in its Cartesian form, a fundamental concept in complex number theory. Specifically, we will solve for 227_i_ and represent the result in the a + bi format, where a is the real part and b is the imaginary part. Understanding Cartesian form is essential for performing various operations with complex numbers, such as addition, subtraction, multiplication, and division.

Understanding Complex Numbers and Cartesian Form:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i.e., _i_² = -1). The real part of the complex number is a, and the imaginary part is b.

The Cartesian form, also known as the rectangular form, is the standard way of representing complex numbers. It allows us to visualize complex numbers on a complex plane, where the horizontal axis represents the real part (a) and the vertical axis represents the imaginary part (b). Each complex number a + bi corresponds to a unique point (a, b) on the complex plane.

Solving for 227i and Representing in Cartesian Form:

The given expression is 227_i_. This can be interpreted as 227 multiplied by the imaginary unit i. To express this in Cartesian form, we need to identify the real and imaginary parts.

Notice that there is no real part explicitly present in the expression 227_i_. This means that the real part is 0. The imaginary part is the coefficient of i, which is 227.

Therefore, we can write 227_i_ in Cartesian form as:

0 + 227_i_

This is the Cartesian form representation of the complex number 227_i_. The real part is 0, and the imaginary part is 227. On the complex plane, this number would be represented by the point (0, 227), lying on the positive imaginary axis.

Importance and Applications of Cartesian Form:

The Cartesian form is crucial for performing arithmetic operations with complex numbers. Addition and subtraction are straightforward: we simply add or subtract the real parts and the imaginary parts separately.

For example:

(a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) - (c + di) = (a - c) + (b - d)i

Multiplication and division are a bit more involved but still rely on the Cartesian form as a foundation.

Furthermore, the Cartesian form provides a clear geometric interpretation of complex numbers, allowing us to visualize them as points on a plane. This geometric representation is essential for understanding concepts such as the magnitude and argument of a complex number.

In various fields such as electrical engineering, physics, and signal processing, complex numbers are used extensively. Representing complex numbers in Cartesian form is often the first step in solving problems and performing calculations in these domains. The ability to convert between different forms of complex numbers, including Cartesian form, is a valuable skill for anyone working with complex numbers.

By understanding the Cartesian form and its significance, we can effectively manipulate and interpret complex numbers, unlocking their power in solving a wide range of problems.

Conclusion

In summary, we've explored two distinct mathematical problems: finding the third side of a triangle using the Law of Cosines and expressing a complex number in Cartesian form. The Law of Cosines proves essential for solving triangles when two sides and the included angle are known, demonstrating its practical application in fields like surveying and navigation. Converting complex numbers into Cartesian form provides a clear representation and facilitates arithmetic operations, crucial in areas such as electrical engineering and physics. Both concepts highlight the importance of mathematical tools in solving real-world problems across various disciplines.