Problem 6 - Precalculus

Question

You are given the parametric equations of a curve and a value for the parameter \(t\). Find the coordinates of the point on the curve corresponding to the given value of \(t\). $$x=\sin t-\sin 2 t, y=\cos t+\cos 2 t ; t=2 \pi / 3$$

Step-by-Step Solution

Answer

The coordinates are \((\sqrt{3}, -1)\).

Step 1: Understand the Problem

We need to find the coordinates \((x, y)\) of a point on the curve given by the parametric equations \(x = \sin t - \sin 2t\) and \(y = \cos t + \cos 2t\), for the specific value \(t = \frac{2\pi}{3}\).

Step 2: Substitute the Parameter in x Equation

Substitute \(t = \frac{2\pi}{3}\) into the equation for \(x\): \[ x = \sin\left(\frac{2\pi}{3}\right) - \sin\left(2 \times \frac{2\pi}{3}\right) \] = \( \sin\left(\frac{2\pi}{3}\right) - \sin\left(\frac{4\pi}{3}\right)\).

Step 3: Simplify x Equation

Evaluate \(\sin\left(\frac{2\pi}{3}\right)\) and \(\sin\left(\frac{4\pi}{3}\right)\):\[\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\]. So, \(x = \frac{\sqrt{3}}{2} - (-\frac{\sqrt{3}}{2}) = \sqrt{3}\).

Step 4: Substitute the Parameter in y Equation

Substitute \(t = \frac{2\pi}{3}\) into the equation for \(y\): \[ y = \cos\left(\frac{2\pi}{3}\right) + \cos\left(2 \times \frac{2\pi}{3}\right) \] = \( \cos\left(\frac{2\pi}{3}\right) + \cos\left(\frac{4\pi}{3}\right)\).

Step 5: Simplify y Equation

Evaluate \(\cos\left(\frac{2\pi}{3}\right)\) and \(\cos\left(\frac{4\pi}{3}\right)\):\[\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}, \quad \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}\]. So, \(y = -\frac{1}{2} + (-\frac{1}{2}) = -1\).

Step 6: Provide the Solution

The coordinates of the point on the curve corresponding to \(t = \frac{2\pi}{3}\) are \((x, y) = (\sqrt{3}, -1)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and evaluating expressions involving trigonometric functions like sine and cosine. These identities help us understand the behavior of angles and their respective trigonometric values.
For example, knowing that

• \(\sin(\pi - \theta) = \sin(\theta)\) and \(\cos(\pi - \theta) = -\cos(\theta)\)
• \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
• and \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)

can be particularly helpful.
This is because they allow us to evaluate expressions where the angles are multiples or combinations of simpler angles, like in our original problem.
In the given exercise, we use these identities to simplify and find numerical values for functions like \(\sin\left(\frac{2\pi}{3}\right)\) and \(\sin\left(\frac{4\pi}{3}\right)\). Knowing these identities helps make trigonometry more intuitive, and assists in solving parametric equations efficiently.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate system to interpret and solve geometric problems. It connects algebra and geometry through graphing equations on coordinate planes.
In the context of parametric equations, the coordinate geometry allows us to map positions in a plane using parameters instead of a direct relationship between \(x\) and \(y\).

• Parametric equations provide a set of coordinates \((x, y)\) based on a third variable, often denoted as \(t\), called the parameter.
• By providing a specific \(t\), as given \(t = \frac{2\pi}{3}\), we can find a specific point on the curve by calculating both \(x\) and \(y\) using their respective equations.

This approach allows for describing more complex curves that are otherwise difficult to express in a single equation form, such as loops or waves, that are often found in real-world applications.

Curve Analysis

Curve analysis involves studying the shape or path defined by parametric equations, often involving different mathematical techniques to understand its properties.
By analyzing a curve, you can find positions and behaviors that are significant, like points of intersection, maximums, minimums, and periodic behavior.

• When given parametric equations, to analyze a curve, one must find specific points by using the parameter \(t\).
• For instance, at \(t = \frac{2\pi}{3}\), we explore the behavior and position of the curve by calculating \(x\) and \(y\) values.

Curve analysis will often involve both algebraic manipulation and geometric interpretation to fully understand the curve's nature.
It is a crucial part of understanding the full implications of parametric representation and is often used in modeling real-world phenomena.