Problem 18 - Algebra 2 and Trigonometry

Question

Find the period of each function. \(y=\cos 0.75 x\)

Step-by-Step Solution

Answer

The period of the function is \(\frac{8\pi}{3}\).

Step 1: Understand the Cosine Function

The standard form of a cosine function is given by \(y = \cos(Bx)\), where \(B\) affects the period of the function.

Step 2: Identify the Value of B

For the function \(y = \cos(0.75x)\), the value of \(B\) is 0.75.

Step 3: Use the Period Formula

The period \(P\) of a cosine function \(y = \cos(Bx)\) is given by the formula \(P = \frac{2\pi}{B}\).

Step 4: Calculate the Period

Substitute \(B = 0.75\) into the period formula to find the period: \[P = \frac{2\pi}{0.75} = \frac{2\pi}{\frac{3}{4}} = \frac{2\pi \times 4}{3} = \frac{8\pi}{3}\]

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions often abbreviated as "cos." In a Cartesian coordinate system, the cosine of an angle can be visualized as the x-coordinate of the point on the unit circle corresponding to that angle. This function outputs values oscillating between -1 and 1.

Key features of the cosine function include:

• It is an even function, meaning it satisfies the property that \({\text{cos}(-x) = \text{cos}(x)}\).
• Its graph is a wave, repeating every full cycle, traditionally starting at (1,0) when the angle is zero.
• The basic cosine graph undulates above and below the x-axis, peaking at 1 and troughing at -1.

Understanding how the cosine function behaves in different scenarios is essential for solving trigonometric problems. For example, when a coefficient \(B\) multiplies the angle \(x\) in \(\cos(Bx)\), it affects the frequency and period of the function.

Period of a Function

The period of a function refers to the interval over which the function completes one full cycle before repeating its pattern. This concept is crucial in trigonometry, especially in sine and cosine functions. The standard cosine and sine functions have a period of \(2\pi\), meaning they repeat their values every \(2\pi\) radians.

When a function is altered to \(y = \cos(Bx)\), the coefficient \(B\) directly modifies the period. The formula to determine the new period is \(P = \frac{2\pi}{B}\).

Let's break it down with an example:

• Given \(y = \cos(0.75x)\), the \(B\) value here is 0.75.
• Using the formula, \(P = \frac{2\pi}{0.75}\), we find that the period becomes \(\frac{8\pi}{3}\).

This concept allows us to determine how often the cosine function repeats over its domain, which is pivotal to many applications where periodicity is involved.

Trigonometric Equations

Trigonometric equations involve trig functions like sine, cosine, and tangent, set to equal a certain value. Solving these requires understanding the fundamental properties and transformations of trigonometric functions.

Let's consider why finding the period is essential in solving some trigonometric equations:

• Identifying the period aids in predicting the behavior of the equation over different intervals.
• With periodic functions, solutions often have an infinite number of solutions at regular intervals, due to the repeating nature of the function.
• By understanding the period, one can write solutions in general forms, such as \(x = A + nP\) for an equation like \(\cos(0.75x) = 0\), where \(A\) is a specific solution, and \(n\) is an integer indicating multiples of the period \(P\).

Being proficient with these equations requires combining algebraic skills with understanding trigonometric graphs and cycles. This comes into play heavily in fields like engineering, physics, and other sciences.