Question

The versed sine and the coversed sine are defined as follows : $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ $$ u=\tan \left(\frac{\pi}{4}-\frac{x}{2}\right) $$

Short answer

Question: Find the relationship between the versed sine (vers x), coversed sine (covers x), and the function u, given that \(u = \tan (\frac{\pi}{4} - \frac{x}{2})\). Answer: The relationship between vers x, covers x, and u is given by: $$ \text{vers } x = \frac{u^2}{1 + u^2} \\ \text{covers } x = 1 - \frac{2u^2}{1 + u^2} $$

Step-by-step solution

Find expressions for sin x and cos x in terms of u

To find sin x and cos x in terms of u, we can use the fact that \( u = \tan (\frac{\pi}{4} - \frac{x}{2} ) \). Let's define a new variable y, such that \( y = \frac{\pi}{4} - \frac{x}{2} \), which means \( x = 2(\frac{\pi}{4} - y) \). We know that \(\tan(y) = u\), now we need to find sin x and cos x in terms of y. Using double angle formulae, we can write: $$ \sin x = \sin 2y = 2\sin y \cos y \\ \cos x = \cos 2y = \cos^2 y - \sin^2 y $$ Now, we can use the identities: \( \sin y = \frac{u}{\sqrt{1 + u^2}} \) and \( \cos y = \frac{1}{\sqrt{1 + u^2}} \), which can be obtained from the definition of the tangent function. By substituting these identities in our expressions for sin x and cos x, we get: $$ \sin x = 2\frac{u}{\sqrt{1 + u^2}}\frac{1}{\sqrt{1 + u^2}} \\ \cos x = \frac{1}{1 + u^2} - \frac{u^2}{1 + u^2} $$

Find vers x and covers x in terms of u

Now that we have expressions for sin x and cos x in terms of u, we can substitute these expressions into the definitions of versed sine and coversed sine. Using the expressions for sin x and cos x, we get: $$ \text{vers } x = 1 - \cos x \\ = 1 - ( \frac{1}{1 + u^2} - \frac{u^2}{1 + u^2}) \\ = \frac{u^2}{1 + u^2} \\ \\ \text{covers } x = 1 - \sin x \\ = 1 - 2\frac{u}{\sqrt{1 + u^2}}\frac{1}{\sqrt{1 + u^2}} \\ = 1 - \frac{2u^2}{1 + u^2} $$ So we have found the relationship between vers x, covers x, and u: $$ \text{vers } x = \frac{u^2}{1 + u^2} \\ \text{covers } x = 1 - \frac{2u^2}{1 + u^2} $$

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They are used to simplify expressions, solve trigonometric equations, and compute specific angles. One of the most fundamental identities are the Pythagorean identities, for example, \(\sin^2 x + \cos^2 x = 1\). These identities are often employed to express one trigonometric function entirely in terms of another, making them invaluable in complex calculations.

When dealing with the versed sine \(\text{vers } x\) and the coversed sine \(\text{covers } x\), we apply these identities to rewrite the expressions in terms of basic trigonometric functions like sine and cosine. This manipulation is crucial for simplifying the problem and finding a relation in terms of the variable \(u\), stretching the utility of identities further to alternate functions such as the versed and coversed sine.

Double Angle Formula

The double angle formulas are a specific set of trigonometric identities that express trigonometric functions of twice an angle in terms of functions of the single angle. These formulas include \(\sin (2x) = 2\sin x\cos x\) and \(\cos (2x) = \cos^2 x - \sin^2 x\), or the alternative forms \(\cos (2x) = 2\cos^2 x - 1\) and \(\cos (2x) = 1 - 2\sin^2 x\). In the context of solving for the versed sine and coversed sine, the double angle formula allows us to express \(\sin x\) and \(\cos x\) in terms of the auxiliary variable \(y\), which simplifies to expressions involving \(u\).

This formula is indispensable when the variable \(x\) is part of a compound angle or when we need to relate it to half-angles, as seen in the exercise. Understanding this concept is key to find the link between the tangent function of a half-angle and the versed and coversed sine, showing the interconnectedness of trigonometric functions.

Tangent Function

The tangent function, often abbreviated as \(\tan\), is one of the primary trigonometric functions and is defined as the ratio of the sine to the cosine of a given angle, \(\tan x = \frac{\sin x}{\cos x}\). When the angle is \(\frac{\pi}{4} - \frac{x}{2}\), as is the case in the exercise, this definition plays a pivotal role in the problem-solving process.

By expressing the tangent of an angle in terms of a variable \(u\), we can then deduce the sine and cosine of the related half-angle through certain trigonometric identities. This approach provides a connection between \(u\) and the functions of the original variable \(x\), allowing the completion of the problem. Understanding the tangent function, and its relation to sine and cosine, is critical because it involves the Pythagorean identity, which is foundational in trigonometry and enables the derivation of various trigonometric expressions.