Distance Between Points (a Cos Θ + B Sin Θ, 0) And (0, A Sin Θ - B Cos Θ)

by ADMIN 74 views

Finding the distance between two points in a coordinate plane is a fundamental concept in mathematics, particularly in coordinate geometry. This article delves into the process of calculating the distance between the points (aextcosθ+bextsinθ,0)(a ext{cos } θ + b ext{sin } θ, 0) and (0,aextsinθ−bextcosθ)(0, a ext{sin } θ - b ext{cos } θ). We will explore the underlying principles, the application of the distance formula, and the algebraic manipulations required to arrive at the solution. Understanding this concept is crucial for various mathematical applications and provides a solid foundation for more advanced topics.

Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem and is used to calculate the distance between two points in a coordinate plane. If we have two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the distance d between them is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

This formula essentially calculates the length of the hypotenuse of a right-angled triangle formed by the horizontal and vertical differences between the coordinates of the two points. The horizontal difference is given by ∣x2−x1∣|x_2 - x_1|, and the vertical difference is given by ∣y2−y1∣|y_2 - y_1|. Squaring these differences, adding them, and then taking the square root gives us the distance between the points.

In our case, the points are (aextcosθ+bextsinθ,0)(a ext{cos } θ + b ext{sin } θ, 0) and (0,aextsinθ−bextcosθ)(0, a ext{sin } θ - b ext{cos } θ). Let's identify the coordinates:

  • x1=aextcosθ+bextsinθx_1 = a ext{cos } θ + b ext{sin } θ
  • y1=0y_1 = 0
  • x2=0x_2 = 0
  • y2=aextsinθ−bextcosθy_2 = a ext{sin } θ - b ext{cos } θ

Now, we will substitute these values into the distance formula and simplify the expression to find the distance between the points. This process involves careful algebraic manipulation and the use of trigonometric identities to arrive at the final result. The application of the distance formula is not just a mathematical exercise; it's a powerful tool used in various fields, including physics, engineering, and computer graphics, to calculate distances and spatial relationships.

Applying the Distance Formula to the Given Points

Now that we have the distance formula and the coordinates of our points, let's substitute the values into the formula. We have:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Substituting the coordinates (aextcosθ+bextsinθ,0)(a ext{cos } θ + b ext{sin } θ, 0) and (0,aextsinθ−bextcosθ)(0, a ext{sin } θ - b ext{cos } θ), we get:

d=[(0−(aextcosθ+bextsinθ))2+((aextsinθ−bextcosθ)−0)2]d = \sqrt{[(0 - (a ext{cos } θ + b ext{sin } θ))^2 + ((a ext{sin } θ - b ext{cos } θ) - 0)^2]}

This simplifies to:

d=[−(aextcosθ+bextsinθ)]2+(aextsinθ−bextcosθ)2d = \sqrt{[-(a ext{cos } θ + b ext{sin } θ)]^2 + (a ext{sin } θ - b ext{cos } θ)^2}

Squaring the terms inside the square root, we get:

d=(aextcosθ+bextsinθ)2+(aextsinθ−bextcosθ)2d = \sqrt{(a ext{cos } θ + b ext{sin } θ)^2 + (a ext{sin } θ - b ext{cos } θ)^2}

Now, we need to expand the squared terms. Recall that (x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2 and (x−y)2=x2−2xy+y2(x - y)^2 = x^2 - 2xy + y^2. Applying these identities, we have:

d=(a2extcos2θ+2abextcosθextsinθ+b2extsin2θ)+(a2extsin2θ−2abextsinθextcosθ+b2extcos2θ)d = \sqrt{(a^2 ext{cos}^2 θ + 2ab ext{cos } θ ext{sin } θ + b^2 ext{sin}^2 θ) + (a^2 ext{sin}^2 θ - 2ab ext{sin } θ ext{cos } θ + b^2 ext{cos}^2 θ)}

Next, we will simplify the expression by combining like terms and using trigonometric identities. This step is crucial for arriving at the most simplified form of the distance. The process of applying the distance formula and simplifying the expression demonstrates the power of algebraic manipulation and the importance of understanding fundamental mathematical principles. This calculation is not just a theoretical exercise; it has practical applications in various fields where distances need to be accurately determined.

Simplifying the Expression and Arriving at the Solution

In the previous section, we expanded the squared terms and obtained the expression:

d=(a2extcos2θ+2abextcosθextsinθ+b2extsin2θ)+(a2extsin2θ−2abextsinθextcosθ+b2extcos2θ)d = \sqrt{(a^2 ext{cos}^2 θ + 2ab ext{cos } θ ext{sin } θ + b^2 ext{sin}^2 θ) + (a^2 ext{sin}^2 θ - 2ab ext{sin } θ ext{cos } θ + b^2 ext{cos}^2 θ)}

Now, let's combine the like terms. Notice that we have 2abextcosθextsinθ2ab ext{cos } θ ext{sin } θ and −2abextsinθextcosθ-2ab ext{sin } θ ext{cos } θ, which cancel each other out. This simplifies the expression to:

d=a2extcos2θ+b2extsin2θ+a2extsin2θ+b2extcos2θd = \sqrt{a^2 ext{cos}^2 θ + b^2 ext{sin}^2 θ + a^2 ext{sin}^2 θ + b^2 ext{cos}^2 θ}

Next, we can rearrange the terms to group the a2a^2 terms and the b2b^2 terms together:

d=a2(extcos2θ+extsin2θ)+b2(extsin2θ+extcos2θ)d = \sqrt{a^2 ( ext{cos}^2 θ + ext{sin}^2 θ) + b^2 ( ext{sin}^2 θ + ext{cos}^2 θ)}

Here, we can use the fundamental trigonometric identity $ ext{sin}^2 θ + ext{cos}^2 θ = 1$. Applying this identity, we get:

d=a2(1)+b2(1)d = \sqrt{a^2 (1) + b^2 (1)}

This simplifies to:

d=a2+b2d = \sqrt{a^2 + b^2}

Therefore, the distance between the points (aextcosθ+bextsinθ,0)(a ext{cos } θ + b ext{sin } θ, 0) and (0,aextsinθ−bextcosθ)(0, a ext{sin } θ - b ext{cos } θ) is a2+b2\sqrt{a^2 + b^2}. This result is independent of the angle θ, which means that the distance remains constant regardless of the value of θ. This simplification demonstrates the elegance of mathematical problem-solving, where seemingly complex expressions can be reduced to simple forms through the application of fundamental principles and identities. This final answer not only solves the specific problem but also provides insight into the geometric relationship between the points.

Conclusion

In summary, we have successfully calculated the distance between the points (aextcosθ+bextsinθ,0)(a ext{cos } θ + b ext{sin } θ, 0) and (0,aextsinθ−bextcosθ)(0, a ext{sin } θ - b ext{cos } θ) using the distance formula. The key steps involved:

  1. Applying the distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
  2. Substituting the coordinates of the given points into the formula.
  3. Expanding the squared terms and simplifying the expression.
  4. Using the trigonometric identity $ ext{sin}^2 θ + ext{cos}^2 θ = 1$ to further simplify the expression.
  5. Arriving at the final answer: d=a2+b2d = \sqrt{a^2 + b^2}.

The final result, a2+b2\sqrt{a^2 + b^2}, shows that the distance between the two points is independent of the angle θ. This means that the distance remains constant regardless of the value of θ. This problem illustrates the importance of understanding and applying fundamental mathematical concepts such as the distance formula and trigonometric identities. It also highlights the power of algebraic manipulation in simplifying complex expressions.

This type of problem is not only a valuable exercise in mathematics but also has practical applications in various fields such as physics, engineering, and computer graphics. Understanding how to calculate distances between points is crucial for many real-world applications, and this example provides a clear and concise demonstration of the process. By mastering these concepts, students can build a strong foundation for more advanced mathematical studies and practical problem-solving.