Question

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

Short answer

Based on the given expression for "u" and the definitions of versed and coversed sine functions, the final expression for "u" in terms of these trigonometric functions is: $$ u=\frac{\text{covers} \frac{x}{1-x}}{\text{vers} \frac{x}{1-x}} $$

Step-by-step solution

Express the tan function as sin/cos

We can start by rewriting the "u" expression as the ratio of sine and cosine: $$ u=\tan\frac{x}{1-x}=\frac{\sin \frac{x}{1-x}}{\cos\frac{x}{1-x}} $$

Utilize sin^2(x) + cos^2(x) = 1

We can now use the Pythagorean trigonometric identity sin^2(x) + cos^2(x) = 1 to express sine or cosine in terms of the other. First, we'll rewrite the sine expression in terms of cosine, and vice versa: $$ \sin^2\frac{x}{1-x} = 1 - \cos^2\frac{x}{1-x}\\ \cos^2\frac{x}{1-x} = 1 - \sin^2\frac{x}{1-x} $$

Introduce vers and covers notation

Next, we can replace sine and cosine expressions with their corresponding versed sine and coversed sine functions, according to their given definitions. This will help us use the information given in the exercise: $$ 1 - \cos\frac{x}{1-x}=1 - \text{vers}\frac{x}{1-x}\\ 1 - \sin\frac{x}{1-x}=1 - \text{covers}\frac{x}{1-x} $$

Solve for sin and cos in terms of vers and covers

Isolate the sine and cosine expressions in each equation: $$ \cos\frac{x}{1-x} = \text{vers}\frac{x}{1-x}\\ \sin\frac{x}{1-x} = \text{covers}\frac{x}{1-x} $$

Substitute expressions for sin and cos into the u formula

Now, we can substitute our expressions for sine and cosine from Step 4 into our expression for u from Step 1: $$ u=\frac{\text{covers}\frac{x}{1-x}}{\text{vers}\frac{x}{1-x}} $$

Simplify u expression

Finally, we can simplify the expression for u: $$ u=\frac{\text{covers} \frac{x}{1-x}}{\text{vers} \frac{x}{1-x}} $$ This is our final expression for u in terms of versed sine and coversed sine functions.

Key concepts

Versed Sine

The concept of the versed sine, often abbreviated as "versin", is not as commonly used today as the basic trigonometric functions, but it carries historical significance and certain practical applications. The versed sine of an angle \(x\) is defined as:

• \( \text{vers } x = 1 - \cos x \)

This means that versed sine is simply one minus the cosine of the angle. It represents the vertical distance from the top of the unit circle to the cosine of the angle.
This distance grows from zero at \( x = 0 \) to two at \( x = \pi \), reflecting the elegance in its symmetry and how it relates to the cosine curve.
The versed sine is utilized in various fields of science and engineering, where the complementary aspects of sine and cosine are beneficial. Understanding the versed sine allows you to see how trigonometric identities extend beyond the primary functions of sine, cosine, and tangent, and offers a broader perspective of angles and their properties.

Coversed Sine

Just like the versed sine, the coversed sine is another supplementary trigonometric function, which is sometimes referred to in its abbreviated form as "coversin". The coversed sine of an angle \(x\) is given by:

• \( \text{covers } x = 1 - \sin x \)

This expression represents a simple operation: subtracting the sine of the angle from one. It can be visualized as the horizontal distance from the tips of the sine wave (at \( \sin x = 1 \)) to the top of the unit circle on the same horizontal line.
Just like its counterpart, the coversed sine is rarely used in modern curriculums, but holds valuable understanding in specific practical applications, similar to instances involving pendulum motion in physics or certain types of waveforms.
Both the versed and coversed sines help form a comprehensive framework for studying trigonometry, offering tools to manipulate and solve various trigonometric equations where standard sine or cosine forms may be cumbersome.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry that stems from the Pythagorean theorem. It is one of the most widely known trigonometric identities and is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This equation illustrates that for any angle \(x\), the sum of the squares of its sine and cosine values equals one. This identity is a crucial tool in trigonometry because it allows you to express one trigonometric function in terms of another.
The identity can be modified to obtain different trigonometric relationships. For example, it can be rearranged to solve for sine or cosine:

• \( \sin^2 x = 1 - \cos^2 x \)
• \( \cos^2 x = 1 - \sin^2 x \)

These variations were utilized in solving the given exercise. By expressing sine and cosine in terms of conversed or reversed sines, the identity facilitates the simplification of many trigonometric expressions.
Having a good grasp of the Pythagorean identity allows for a smoother navigation through the complexities of trigonometry, making it an essential part of the foundation for anyone studying mathematics.