Question

a) Graph the function \(f(x)=\cos \left(x-\frac{\pi}{2}\right)\). b) Consider the graph. Write an equation of the function in the form \(y=a \sin b(x-c)+d\). c) What conclusions can you make about the relationship between the two equations of the function?

Short answer

The function \(f(x) = \, \cos \left(x - \frac{\pi}{2} \right)\) is equivalent to \(y = \, \sin(x)\). The relationship shows that a horizontal shift of \( \frac{\pi}{2} \) in the cosine function is the sine function.

Step-by-step solution

Graph the Function

To graph the function \(f(x) = \, \cos \left(x - \frac{\pi}{2} \right)\), first note that the cosine function is shifted. The standard cosine graph is horizontally shifted to the right by \( \frac{\pi}{2} \). Start by drawing the standard cosine wave which has a period of \(2\pi\), and then shift each point of the graph to the right by \( \frac{\pi}{2} \).

Identify the Sine Function Form of the Cosine

Recall the trigonometric identity: \( \cos(x - \frac{\pi}{2}) = \sin(x) \). Therefore, the function \(f(x) = \cos \left(x - \frac{\pi}{2} \right)\) can be rewritten as \(y = \, \sin(x)\).

Write the Equation in the Sine Form

Given the identified function from step 2, the cosine function in the sine form is: \(y = \sin(x)\). This matches the form \(y = a \sin\, b(x - c) + d\) where \(a = 1\), \(b = 1\), \(c = 0\), and \(d = 0\). Hence, the sine form of the cosine function is simply: \(y = \sin(x)\).

Analyze the Relationship

By comparing the original function \(f(x) = \, \cos \left(x - \frac{\pi}{2} \right)\) and its sine form \(y = \, \sin(x)\), we conclude that the sine function and the horizontally shifted cosine function are equivalent due to their trigonometric identity relationship. This results in the cosine function being expressed as a sine function.

Key concepts

Cosine Function

The **cosine function**, symbolized as \(\cos(x)\), demonstrates the relationship between the angle and the adjacent and hypotenuse sides of a right-angled triangle. When graphing **\(f(x) = \, \cos \, x\)**, the result is a smooth, wave-like curve repeating every \(2\pi\) radians. This cycle, known as the period, exhibits maximum values of 1 and minimum values of -1 regularly. With transformations, including shifts and stretches, the cosine function can display various forms, like \(f(x) = \, \cos \, \, (x - \frac{\pi}{2})\). Shifting the graph to the right by \frac{\pi}{2}\ is an example of a horizontal **transformation**, affecting its placement without altering the wave pattern.

Sine Function

The **sine function**, denoted as \(\text{sin}(x)\), is another fundamental trigonometric function representing the y-coordinate of a point on the unit circle as the angle varies. For the standard **sine function**, \(f(x) = \, \sin \, x\), the graph also depicts a wave-like curve repeating every \(2\pi\) radians. Identical to the cosine graph in **shape**, the sine function peaks at 1 and dips at -1. Importantly, there are relationships between sine and cosine functions, such as \(\text{cos}(x \, - \, \, \frac{\pi}{2}) \, = \, \text{sin}(x)\). By understanding this identity, a transformation of the cosine graph can be expressed using the sine function, offering different perspectives for solving problems.

Trigonometric Identity

A **trigonometric identity** is an equation that holds true for all permitted values of the variables involved. One crucial identity is \- \(\text{cos}(x \, - \, \frac{\pi}{2}) \, = \, \text{sin}(x)\) -\. It suggests that shifting the cosine function to the right by \(\frac{\pi}{2}\) radians yields the sine function. Trigonometric identities **simplify** complex expressions and make it easier to solve trigonometric equations. Additionally, they play a significant role in calculus and higher mathematics. Mastering these identities allows one to interchange between functions and aids in graph transformations and integrations.

Function Transformation

In both trigonometry and broader mathematical contexts, **function transformation** involves modifying the appearance of a graph without altering its core properties. Transformations can include:

• **Shifting**: Moving the entire graph horizontally or vertically without reshaping it. For instance, \(f(x) = \, \cos \, (x \, - \, \frac{\pi}{2})\) results in a horizontal shift of the cosine function.
• **Stretching/Compressing**: Altering the amplitude or period of the graph. For example, changing the coefficient before \(\text{cos}\) or \(\text{sin}\) affects the height (amplitude) of the waves.
• **Reflecting**: Flipping the graph over a given axis. For example, \(f(x) = -\text{cos}(x)\) reflects over the x-axis.

Understanding transformations helps in visualizing and solving problems involving trigonometric functions, as seen in the given cosine function transformed to its sine form.