Finding Zeros Of Polar Equation R = 2sin(5θ) A Step By Step Guide

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Determining the zeros of polar equations like r = 2sin(5θ) involves finding the values of θ for which the radial distance r is equal to zero. This exploration delves into the intricacies of solving such equations, providing a step-by-step methodology, and offering a clear understanding of the underlying principles. In this comprehensive guide, we will meticulously walk through the process of finding the zeros of r = 2sin(5θ), ensuring that you grasp not just the how, but also the why behind each step. This understanding is crucial for tackling similar problems and for gaining a deeper appreciation of polar equations and their graphical representations. Whether you're a student grappling with calculus or an enthusiast eager to expand your mathematical toolkit, this guide will provide you with the knowledge and confidence to master the art of finding zeros in polar coordinates.

Understanding Polar Equations and Zeros

Polar equations provide an alternative way to represent curves and shapes, distinct from the familiar Cartesian coordinate system. Instead of defining points using x and y coordinates, polar coordinates use a radial distance r from the origin and an angle θ from the positive x-axis. This system proves particularly useful for describing curves with radial symmetry, such as circles, spirals, and rose curves. The equation r = 2sin(5θ) is a classic example of a polar equation, representing a rose curve with five petals. Understanding the nature of these equations is paramount to grasping how their zeros are determined.

A zero of a polar equation is a value of θ for which r = 0. Geometrically, these zeros correspond to points where the curve passes through the origin. In the context of the equation r = 2sin(5θ), finding the zeros means identifying the angles θ at which the rose curve intersects the origin. These points are critical in sketching the graph of the curve and understanding its overall shape and behavior. The zeros essentially act as anchors, defining the starting and ending points of the petals of the rose curve. Therefore, accurately determining these zeros is a fundamental step in analyzing and visualizing the polar equation.

Step-by-Step Solution for r = 2sin(5θ)

To determine the zeros of the polar equation r = 2sin(5θ), we need to solve the equation 2sin(5θ) = 0. This is achieved by following a series of logical steps, each building upon the previous one. The process involves understanding the properties of the sine function and applying algebraic techniques to isolate the variable θ. This methodical approach not only leads to the correct solution but also enhances your problem-solving skills in trigonometry and polar coordinates.

1. Setting the Equation to Zero

The first step is to set the equation equal to zero: 2sin(5θ) = 0. This step is the foundation of the entire solution, as it transforms the problem into a standard trigonometric equation. By setting the radial distance r to zero, we are essentially looking for the angles θ at which the curve intersects the origin. This is a direct application of the definition of a zero in the context of polar equations. The subsequent steps will build upon this foundation, using trigonometric identities and algebraic manipulations to isolate θ and find its values.

2. Isolating the Sine Function

Next, we isolate the sine function by dividing both sides of the equation by 2: sin(5θ) = 0. This simplification is crucial because it allows us to focus solely on the trigonometric part of the equation. By isolating sin(5θ), we can directly apply our knowledge of the sine function's behavior and its zeros. This step highlights the importance of algebraic manipulation in solving trigonometric equations. The simplified equation now clearly indicates that we need to find the angles for which the sine of 5θ is zero.

3. Finding General Solutions for 5θ

The sine function is zero at integer multiples of π. Therefore, we can write the general solution for 5θ as: 5θ = nπ, where n is an integer. This step leverages the periodic nature of the sine function, acknowledging that sin(x) = 0 at x = 0, π, 2π, -π, and so on. By expressing the solution in terms of n, we capture all possible angles for which sin(5θ) equals zero. This general solution is a crucial stepping stone towards finding the specific values of θ that satisfy the original equation.

4. Solving for θ

To find the values of θ, we divide both sides of the equation by 5: θ = (nπ)/5, where n is an integer. This step isolates θ, providing a general formula for all possible solutions. The formula θ = (nπ)/5 reveals that the zeros occur at angles that are integer multiples of π/5. This is a significant insight, as it allows us to generate a series of solutions by simply substituting different integer values for n. This step demonstrates the power of algebraic manipulation in solving for the unknown variable.

5. Identifying Solutions within the Interval [0, 2π)

We need to find solutions for θ within the interval [0, 2π) to cover one full rotation in the polar coordinate system. We do this by substituting integer values for n in the equation θ = (nπ)/5 and checking if the resulting θ falls within the specified interval. This step is crucial for obtaining a complete set of unique solutions within a single period. The interval [0, 2π) is chosen because it represents a full circle in polar coordinates, and any angle outside this range would be coterminal with an angle within the range.

Let's substitute values for n:

  • n = 0: θ = (0π)/5 = 0
  • n = 1: θ = (1π)/5 = π/5
  • n = 2: θ = (2π)/5
  • n = 3: θ = (3π)/5
  • n = 4: θ = (4π)/5
  • n = 5: θ = (5π)/5 = π
  • n = 6: θ = (6π)/5
  • n = 7: θ = (7π)/5
  • n = 8: θ = (8π)/5
  • n = 9: θ = (9π)/5
  • n = 10: θ = (10π)/5 = 2π

We stop at n = 9 because n = 10 gives us θ = 2π, which is the upper bound of our interval but not included (we only consider values strictly less than 2π).

Therefore, the solutions within the interval [0, 2π) are: θ = 0, π/5, (2π)/5, (3π)/5, (4π)/5, π, (6π)/5, (7π)/5, (8π)/5, and (9π)/5.

Analyzing the Solutions

The solutions we found, θ = 0, π/5, (2π)/5, (3π)/5, (4π)/5, π, (6π)/5, (7π)/5, (8π)/5, and (9π)/5, represent the angles at which the rose curve r = 2sin(5θ) intersects the origin. These points are crucial for understanding the shape and symmetry of the curve. The rose curve has five petals, and these zeros mark the points where the petals meet at the origin. Analyzing these solutions provides valuable insights into the graphical representation of the polar equation.

Notice that the solutions are evenly spaced at intervals of π/5. This even spacing is a characteristic feature of rose curves of the form r = asin(nθ) or r = acos(nθ), where n is an integer. The number of petals in the rose curve is determined by the coefficient of θ inside the trigonometric function. In this case, since the coefficient is 5, the curve has five petals. The zeros we found correspond to the angles at which these petals intersect at the origin.

Selecting the Correct Answer

Based on our step-by-step solution, we identified the zeros of r = 2sin(5θ) within the interval [0, 2π) as θ = 0, π/5, (2π)/5, (3π)/5, (4π)/5, π, (6π)/5, (7π)/5, (8π)/5, and (9π)/5. Now, let's compare these solutions with the options provided:

A. θ = 0, π/5, (4π)/5, π B. θ = π/5, (2π)/5, (3π)/5, (4π)/5 C. θ = 0, π/5, (2π)/5, (3π)/5, (4π)/5, π, (6π)/5, (7π)/5, (8π)/5, (9π)/5

Upon careful comparison, we can see that option C contains all the zeros we calculated. Therefore, option C is the correct answer.

The other options are incorrect because they do not include the complete set of zeros for the given polar equation within the specified interval. Option A is missing several zeros, while option B omits the zero at θ = 0 and includes only a subset of the solutions. This highlights the importance of finding all possible solutions within the interval [0, 2π) to accurately determine the zeros of a polar equation.

Conclusion

In conclusion, determining the zeros of the polar equation r = 2sin(5θ) involves a systematic approach that combines trigonometric principles and algebraic techniques. By setting the equation to zero, isolating the sine function, finding general solutions, solving for θ, and identifying solutions within the interval [0, 2π), we successfully found all the angles at which the curve intersects the origin. The correct set of zeros is θ = 0, π/5, (2π)/5, (3π)/5, (4π)/5, π, (6π)/5, (7π)/5, (8π)/5, and (9π)/5, which corresponds to option C.

This detailed exploration not only provides the solution to the specific problem but also equips you with a comprehensive methodology for tackling similar problems involving polar equations. Understanding the zeros of polar equations is crucial for sketching their graphs and analyzing their properties. By mastering these techniques, you can confidently navigate the world of polar coordinates and gain a deeper appreciation for the beauty and versatility of mathematical representations.

Remember, the key to success in mathematics lies in understanding the underlying principles and applying them systematically. Practice is essential for honing your skills and building confidence. So, take on new challenges, explore different polar equations, and continue to expand your mathematical horizons. The journey of mathematical discovery is a rewarding one, and with dedication and perseverance, you can unlock its many wonders.