Question

Find real numbers a and b such that equations \(A\) and \(B\) describe the same curve. $$\begin{aligned}&A: x=10 \sin t, y=10 \cos t ; 0 \leq t \leq 2 \pi\\\&B: x=10 \sin 3 t, y=10 \cos 3 t ; a \leq t \leq b\end{aligned}$$

Short answer

Create a short answer question based on the above step-by-step solution: Question: Given the parametric equations A: x = 10sin(t), y = 10cos(t) for 0 ≤ t ≤ 2π, and B: x = 10sin(3t), y = 10cos(3t) for a ≤ t ≤ b, find the real numbers 'a' and 'b' such that both equations describe the same curve. Answer: a = 0, b = 2π

Step-by-step solution

Analyze given parametric equations for curve A and B

First, we can rewrite both parametric equations as: $$\begin{aligned} A: x = 10 \sin t, y = 10 \cos t ; 0 \leq t \leq 2 \pi \newline B: x = 10 \sin 3t, y = 10 \cos 3t ; a \leq t \leq b \end{aligned}$$ Notice that for equation A, x and y are sinusoidal functions with a period of \(2 \pi\). Therefore, the entire curve A is described within the interval \(0 \leq t \leq 2 \pi\). For equation B, we have three times the angle in the sinusoidal functions, which means the period of these functions would be \(\frac{2 \pi}{3}\). We need to find the interval \(a \leq t \leq b\) so that equation B describes the same curve as equation A.

Determine the necessary condition for the same curve

In order for both equations to describe the same curve, they must have the same period. Thus, we need the interval for equation B to cover a period of \(2 \pi\), or in other words: $$b - a = 2 \pi$$

Calculate the number of periods for equation B

We know that a single period of equation B is \(\frac{2 \pi}{3}\). To cover a period of \(2 \pi\) like equation A, we need: $$n \times \frac{2 \pi}{3} = 2 \pi$$ where n is the number of periods of equation B that will cover a period of \(2 \pi\). Solving for n, we get: $$n = \frac{3}{1}$$ So there are 3 periods of equation B that will cover a period of \(2 \pi\) like equation A.

Find the interval for equation B

Now that we know the interval needs to cover 3 periods of equation B, let’s find the interval: We already have: $$b - a = 2 \pi$$ Since the interval needs to be positive, we can assume that a starts at 0: $$b - 0 = 2 \pi$$ Therefore, \(b = 2 \pi\) and \(a = 0\). So the interval for equation B is \(0 \leq t \leq 2 \pi\).

Final Answer

The real numbers a and b for equation B that describe the same curve as equation A are: $$a = 0, b = 2 \pi$$

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental components in mathematics, particularly when dealing with circles and oscillatory motions. These functions include sine, cosine, and tangent functions, among others.

- **Sine and Cosine Functions**: In parametric equations such as those given in the exercise, sine and cosine represent the circular motion in terms of angles and radii. - For instance, in equation A, the function \(x = 10 \sin t\) and \(y = 10 \cos t\) utilize the unit circle concept, where x is the horizontal projection, and y is the vertical projection of a point on a circle with radius 10. - **Understanding Parameters**: Here, the parameter \(t\) represents the angle in radians, which gives the position on the circle as it varies from \(0\) to \(2\pi\), completing one full circle.
Trigonometric functions are inherently periodic, which is the basis of analyzing curves formed by such parametric equations. They help in determining the behavior of curves describing circular motion or waves.

Curve Analysis

Curve analysis involves examining the properties of a curve, such as its shape, size, and how it changes with varying parameters. This analysis often uses parametric equations, as seen in exercises like this one.

- **Parametric Equations**: These describe curves by expressing coordinates as functions of a third variable, commonly denoted as \(t\). In the equation \(x = 10 \sin t\), \(y = 10 \cos t\), both x and y are functions of \(t\), which allows tracing out a curve step by step as \(t\) varies.
- **Same Curve Analysis**: For two parametric equations to represent the same curve, they must cover identical paths. In this exercise, it's achieved by ensuring both equations have parameters that cover the entire cycle of the circle.

Analyzing such parametric curves is essential for understanding how changes in parameters affect the path of the curve. It helps determine whether different sets of equations represent the same geometric path or distinct shapes.

Periodicity

Periodicity is a key concept related to repeating patterns in mathematics. In the context of trigonometric functions and parametric equations, it refers to how often a cycle of a pattern repeats.

- **Period of Trigonometric Functions**: For circle-based functions like sine and cosine, the standard period is \(2\pi\), meaning the functions repeat their values after this interval. However, when angles are modified, as in \(10 \sin 3t\), the period changes accordingly.
- **Finding Periods**: To ensure curves described by parametric equations are the same, they must span the same total period. In the exercise, \(3t\) modifies the period of trigonometric functions from \(2\pi\) to \(\frac{2\pi}{3}\), necessitating a threefold repetition to equate to the unchanged curve A's period.

Understanding periodicity ensures that the comparison of different parametric equations correctly accounts for their cycles. It helps in determining intervals over which identical curve paths are traced, confirmed by equating total cycles rather than individual cycles.