Question

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

Short answer

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

Step-by-step solution

Recall the double-angle formula for cosine

First, recall the double-angle formula for cosine, which states that \(\cos 2\theta = 1 - 2\sin^2 \theta\). We will use this formula to express \(\cos x\) and \(\cos \frac{x}{2}\) in terms of sine.

Express \(\cos x\) and \(\cos \frac{x}{2}\) in terms of sine using the double-angle formula

Using the double-angle formula \(\cos 2\theta = 1 - 2\sin^2 \theta\), we can rewrite \(\cos x\) and \(\cos \frac{x}{2}\) as follows: $$ \cos x = 1 - 2\sin^2 \frac{x}{2} $$ $$ \cos 2\left(\frac{x}{2}\right) = 1 - 2\sin^2 \left(\frac{x}{2}\right) $$ Notice that we now have expressions for \(\cos x\) and \(\cos \frac{x}{2}\) in terms of sine.

Express versed sine and coversed sine in terms of sine using the expressions from Step 2

Now, we can use the expressions from Step 2 to rewrite the given definitions of versed sine and coversed sine in terms of sine: $$ \text{vers } x = 1-\cos x = 1 - \left(1 - 2\sin^2 \frac{x}{2}\right) = 2\sin^2 \frac{x}{2} $$ $$ \text{covers } x =1-\sin x = 1 - \sin x $$

Rewrite the function u in terms of versed sine and coversed sine

Now that we have expressions for versed sine and coversed sine in terms of sine, we will rewrite the given function u as: $$ u = \frac{\cos \frac{x}{2}}{x} $$ We can use the expressions from Step 2 to express \(\cos \frac{x}{2}\) in terms of sine: $$ \cos \frac{x}{2} = 1 - 2\sin^2 \frac{x}{4} $$ Now, we can rewrite the function u as: $$ u = \frac{1 - 2\sin^2 \frac{x}{4}}{x} $$ Finally, we can express the function u in terms of versed sine: $$ u = \frac{\text{vers}\frac{x}{2}}{x} $$ The function u is now expressed in terms of versed sine, as the exercise requires.

Key concepts

Double-angle formulas

Double-angle formulas are equations in trigonometry that express trigonometric functions of double angles in terms of single angles. They are useful for solving various mathematical problems and simplifying complex expressions. One of the most frequently used double-angle formulas is for cosine, which states that:

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

This formula is particularly handy because it allows us to express the cosine of a double angle in terms of the sine of a single angle. Using this formula, we can determine expressions for other cosine and sine components. For instance, to find the expression for \( \cos x \), we set \( \theta = \frac{x}{2} \), so \( \cos x = 1 - 2\sin^2 \frac{x}{2} \).
The double-angle formulas are not limited to cosine. Similar formulas exist for sine and tangent, expanding the versatility of these tools in problem-solving. Understanding and memorizing these formulas can save time and effort when working through trigonometric problems.

Versed sine

The versed sine, often abbreviated as vers, is a less commonly used trigonometric function defined by the equation:

• \( \text{vers} \, x = 1 - \cos x \)

It represents the vertical distance from the point on the unit circle corresponding to an angle \( x \) to the horizontal intercept. This concept might sound complex, but it is essentially the complement of the cosine. To compute the versed sine using the double-angle formula, we have:

• \( \text{vers} \, x = 1 - (1 - 2\sin^2 \frac{x}{2}) = 2\sin^2 \frac{x}{2} \)

This shows that the versed sine can be readily expressed in terms of the sine function, making it easier to handle in calculations. Despite its rarity in modern mathematics, knowing the versed sine helps in understanding historical and specific applications of trigonometry in various scientific fields.

Coversed sine

The coversed sine is another historical trigonometric concept similar to the versed sine. It is defined as:

• \( \text{covers} \, x = 1 - \sin x \)

Unlike the versed sine, the coversed sine is associated directly with the sine function, subtracting the sine value from 1. It provides insight into specific relationships within trigonometry.
Although it may not be commonly used in today's mathematical problems, the coversed sine offers another perspective on sine function transformations. In some computations, it can be beneficial to express functions in terms of coversed sine to simplify expressions or solve particular trigonometric identities.

Sine function

The sine function, abbreviated as sin, is one of the most fundamental trigonometric functions.

• It is defined for an angle \( x \) as the ratio of the length of the opposite side to the hypotenuse in a right triangle.

This function is periodic, with a period of \( 2\pi \), meaning it repeats its values in a cyclical manner. The sine function is also essential in representing waveforms in physics, such as sound or light waves. The famous double-angle formula \( \cos 2\theta = 1 - 2\sin^2 \theta \) relates the sine and cosine functions, showing their interdependent nature.
The sine function plays a crucial role in formulating various mathematical expressions, including the versed and coversed sines discussed previously. Its significance in both pure and applied mathematics cannot be overstated, as it provides a foundation for understanding more complex trigonometric concepts and real-world applications.