Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=\cos t, y=\sin t ; \quad t=\frac{\pi}{4}$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

Step-by-Step Solution

Answer

The matching ordered pair is D: \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \).

Step 1: Identify Parametric Equations

The parametric equations provided are \( x = \cos t \) and \( y = \sin t \). We need to evaluate these equations at the given \( t \) value, which is \( t = \frac{\pi}{4} \).

Step 2: Evaluate \( x \) at \( t = \frac{\pi}{4} \)

Substitute \( t = \frac{\pi}{4} \) into the equation \( x = \cos t \): \[ x = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}. \]

Step 3: Evaluate \( y \) at \( t = \frac{\pi}{4} \)

Substitute \( t = \frac{\pi}{4} \) into the equation \( y = \sin t \): \[ y = \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}. \]

Step 4: Identify Matching Ordered Pair

The calculations show that at \( t = \frac{\pi}{4} \), \( x = \frac{\sqrt{2}}{2} \) and \( y = \frac{\sqrt{2}}{2} \). This matches the ordered pair in Column II, option D: \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \).

Key Concepts

Trigonometric Functions

In mathematics, trigonometric functions are fundamental for relating angles to side lengths in right triangles. Two of the primary trigonometric functions, sine (\( \sin \)) and cosine (\( \cos \)), are frequently used, especially in the context of circular motion and parametric equations. When you think about a circle, \( \sin \) and \( \cos \) help depict points around the circular edge.
These functions are defined as follows:

• Cosine (\( \cos \)): Relates to the adjacent side and hypotenuse in a right triangle. It measures the horizontal distance to a point on a unit circle from the origin.
• Sine (\( \sin \)): Relates to the opposite side and hypotenuse. It measures the vertical distance to a point on the unit circle from the origin.

Since trigonometric functions are crucial in parametric equations, we use \( t \) to indicate angles or time increments. Evaluating \( \cos \) and \( \sin \) for a specific angle helps to find specific coordinates on the circle, which ties directly back to our exercise by matching parametric pairs to coordinates.
For an angle \( t = \frac{\pi}{4} \), both \( \cos \) and \( \sin \) will output \( \frac{\sqrt{2}}{2} \), pinpointing a specific location on the circle's boundary.

Circular Motion

Circular motion is the trajectory of an object that moves along the circumference of a circle. Think about a ferris wheel: its motion can be represented using circular motion. The interesting part is that we can describe this motion using parametric equations.
Parametric equations such as \( x = \cos(t) \) and \( y = \sin(t) \) describe the path of an object moving around a unit circle, a circle with a radius of 1 unit. These equations give us a snapshot of where an object is at any given time, \( t \).
The parameter \( t \) generally represents the angle in radians, or the time elapsed, depending on what you're modeling. In this exercise's context, \( t = \frac{\pi}{4} \) represents an angle of 45 degrees where our object is diagonally positioned, equidistant from both axes with \( \frac{\sqrt{2}}{2} \) as its x and y coordinates.

Evaluating Functions

Evaluating functions is a key mathematical process, where we substitute a specific input, often a number or a variable, into a function to find the outcome. This helps us determine specific points on a graph.
When handling parametric equations like \( x = \cos t \) and \( y = \sin t \), we evaluate by substituting specific \( t \) values to calculate actual coordinates on a circle. For instance, substituting \( t = \frac{\pi}{4} \) into these functions gives us the components of the ordered pair. This process involves basic arithmetic and understanding of trigonometric values, like knowing \( \cos \left(\frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
By evaluating parametric functions in this way, you can match points from a circle to desired pairs or determine an object's position at any given time in its circular path.