Problem 133 - Precalculus Essentials

Question

Use this information to solve. A ball on a spring is pulled 4 inches below its rest position and then released. After t seconds, the balls distance, \(d\), in inches from its rest position is given by $$d=-4 \cos \frac{\pi}{3} t$$ Find all values of \(t\) for which the ball is 2 inches above its rest position.

Step-by-Step Solution

Answer

Therefore, the ball is 2 inches above its rest position when \(t=2+4k\) and \(t=4+4k\), where \(k\) is any integer.

Step 1: Set up the equation

Given that the distance \(d=-4 \cos \frac{\pi}{3} t\) represents the distance of the ball from its rest position at any given time \(t\), when the ball is 2 inches above its rest position \(d=2\). Therefore, we set up the equation as \(2=-4 \cos \frac{\pi}{3} t\).

Step 2: Solve for cos \(\frac{\pi}{3} t\)

The next step involves solving for \(\cos \frac{\pi}{3} t\). Start by dividing both sides of the equation by -4 to isolate \(\cos \frac{\pi}{3} t\). This will give \(\cos \frac{\pi}{3} t=-\frac{1}{2}\).

Step 3: Solve for t

Using the knowledge of the unit circle or cosine function, it is known that \(\cos x = -\frac{1}{2}\) when \(x=\frac{2\pi}{3}\) or \(x=\frac{4\pi}{3}\). As such, solve for \(t\) by equating \(\frac{\pi}{3} t\) to \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\). Solving both separately would give \(t=2\) and \(t=4\) respectively. Since the cosine function is periodic, there will be other solutions to this equation, namely \(t=2+4k\) and \(t=4+4k\), where \(k\) is any integer.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the behavior of periodic phenomena, such as waves, oscillations, and circular motion. These functions, which include sine, cosine, tangent, and their reciprocals, arise from the study of right-angled triangles and have been extended to define relationships within the unit circle.

For instance, the cosine function, which is represented as \( \text{cos}(x) \), correlates with the x-coordinate of a point on the unit circle. Similarly, the sine function corresponds to the y-coordinate. These functions repeat their values in a regular pattern, making them periodic. In the context of the given problem, the motion of a ball on a spring exhibits periodic behavior that can be described using the cosine function.

Unit Circle

The unit circle is a vital concept in trigonometry and serves as a graphical representation of trigonometric functions. Centered at the origin of a coordinate plane, it has a radius of 1 unit. Positions on the unit circle correspond with angle measures, and the coordinates of any point on the circle are given by \( (\text{cos}(θ), \text{sin}(θ)) \) where \( θ \) is the angle in radians measured from the positive x-axis.

The unit circle facilitates the understanding of the periodic nature of trigonometric functions and allows us to find trigonometric values for various angles. This helps to solve equations like the one in our exercise, by visualizing the cosine values and their corresponding angles.

Cosine Function Periodicity

One of the defining properties of the cosine function is its periodicity. The cosine function is periodic with a period of \( 2π \) radians, meaning that the function repeats its values every \( 2π \) radians along the horizontal axis. This periodic nature implies that if you know the value of cosine for any angle, you can predict the values at intervals of the period.

In practical terms, if \( \text{cos}(x) = y \), then it is also true that \( \text{cos}(x + 2πk) = y \), where \( k \) is any integer. The relevance of this periodicity is visible in our exercise, which involves a repeating motion where the ball reaches the same position at regular intervals due to the repeating nature of the cosine function.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all the angles that satisfy the given trigonometric expression. In many cases, this process requires knowledge of the unit circle and the properties of trigonometric functions such as periodicity. When solving trigonometric equations, one must consider not only the principal solution but also all other solutions within the function's period.

For example, in the exercise, by identifying that \( \text{cos}(x) = -\frac{1}{2} \) at specific reference angles and using cosine's periodicity, we can determine all possible solutions for \( t \) given the equation \( -4 \text{cos}\frac{π}{3} t = 2 \). Incorporating the periodic nature of the cosine function gives a set of infinite solutions calculated using the formula \( t_n = t_0 + 2πn \) for \( n \) as any integer, where \( t_0 \) is a specific solution within one period of the cosine function.