Question

The versed sine and the coversed sine are defined as follows : $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ $$ u=x^{2} \cos \pi x $$

Short answer

Question: Express the given function \(u = x^2 \cos{(\pi x)}\) in terms of the versed sine function. Answer: \(u = x^2 - x^2\text{vers}{(\pi x)}\)

Step-by-step solution

Rewrite the expression using cosine and sine functions

To start, let's rewrite the expression \(u = x^2\cos{(\pi x)}\) using cosine and sine functions.$$u = x^2\cos{(\pi x)}$$

Apply the versed and coversed sine definitions

Next, we'll use the given definitions of the versed sine (\(\text{vers}\, x = 1 - \cos{x}\)) and coversed sine (\(\text{covers}\, x = 1 - \sin{x}\)) to rewrite the expression in terms of these functions. First, let's solve the given definitions for \(\cos{x}\) and \(\sin{x}\): $$\cos{x} = 1 - \text{vers}\, x$$ $$\sin{x} = 1 - \text{covers}\, x$$ Now, we'll replace \(\cos{x}\) in the expression for \(u\) with the expression we found in terms of versed sine function: $$u = x^2(1 - \text{vers}{(\pi x)})$$

Simplify the expression

Next, we will simplify the expression by distributing \(x^2\) to the terms inside the parentheses: $$u = x^2(1) - x^2\text{vers}{(\pi x)}$$ Finally, the simplified expression for \(u\) in terms of the versed sine function is: $$u = x^2 - x^2\text{vers}{(\pi x)}$$ Now we have successfully expressed the given function in terms of the versed sine function, using the provided definitions for the versed and coversed sine functions.

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in calculus, since they describe angles and their relationships with angles. The most common trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These functions are used to model

• Various physical phenomena such as waves and oscillations
• Geometrics involving circles and triangles
• Even complex mathematical problems

For example, the cosine function, which is used in the exercise provided, represents the x-coordinate of a point on a unit circle at a given angle. Trigonometric functions have their origin in coordinate geometry and can be expressed as ratios of sides of a right triangle or points on the unit circle. This makes them extremely useful for calculating derivatives and integrals in calculus.

Versed Sine

The versed sine is one of the lesser-known trigonometric functions, yet quite interesting and useful in simplification processes. It is defined as the difference between the number 1 and the cosine of an angle:\(\text{vers}\theta = 1 - \cos\theta \). The versed sine can simplify calculations involving oscillations and waves, where symmetrical properties are leveraged. It's not as popular but still serves as a convenient shortcut in some cases. By using the versed sine, we rephrase problems involving subtraction from one to create new forms that can sometimes make subsequent mathematical operations smoother. In this exercise, using the versed sine allows us to modify the function and explore its derivative in a simplified manner.

Derivative Calculation

Calculating derivatives is a cornerstone of calculus, providing insights into function behavior such as rates of change, slopes of tangents, and curves. To calculate the derivative of a function, we typically follow a sequence of differentiating operations based on known principles like the power rule, the product rule, and the chain rule. For example:

• The power rule involves bringing down the exponent and reducing it by one, useful in derivative functions like \(f(x) = x^2\).
• The product rule says that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

These rules make finding the derivative systematic and methodical. In the context of the exercise, understanding how derivatives act upon trigonometric and polynomial expressions is vital. Once the expression is in terms of versed sine, the derivative calculation forms key steps where every function interaction is clearly laid out.

Function Simplification

Simplification is a crucial process in calculus, aimed at making expressions more manageable. This step can involve factoring, combining like terms, or rewriting expressions using known identities. In our example, function simplification was accomplished by utilizing the properties of the versed sine to transform the original complex expression into an equivalent, but simpler form. By distributing, factoring, or collecting like terms, we remove complexities and reveal intrinsic properties of functions that are easier to differentiate or integrate. Being adept at simplification allows mathematicians and students to handle equations with precision and ease, often reducing large calculations into smaller, more intuitive chunks. It's an essential toolkit for solving calculus problems efficiently.