Question

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

Short answer

#tag_title# Step 6: Determine the domain of the function #tag_content# The domain of a function consists of all the possible input values (x-values) that allow the function to output a real value. In this case, since vers x and covers x are both derived from the sine and cosine functions, the domain of u will be the same as the domain of sine and cosine functions, which is all real numbers. Therefore, the domain of u is: $$\text { Domain }(u) = (-\infty, +\infty)$$ #tag_title# Step 7: Find critical points of the function #tag_content# To find the critical points of the function u, we need to find the points where the derivative of u is either equal to zero or undefined. First, let's find the derivative of u with respect to x: $$\frac{du}{dx} = 2(1 - \text { covers } x) - 2x\frac{d(\text { covers } x)}{dx}(1 - \text { vers } x) - 2x (1 - \text { covers } x)\frac{d(\text { vers } x)}{dx}$$ Since, \(d(\text { covers } x)/dx = -\cos x\) and \(d(\text { vers } x)/dx = \sin x\), the derivative of u becomes: $$\frac{du}{dx} = 2(1 - \text { covers } x) + 2x\cos x(1 - \text { vers } x) - 2x (1 - \text { covers } x)\sin x$$ Now, to find the critical points of u, we set the derivative equal to zero and solve for x: $$2(1 - \text { covers } x) + 2x\cos x(1 - \text { vers } x) - 2x (1 - \text { covers } x)\sin x = 0$$ This equation may not have a simple analytical solution, and numerical methods may be used to find the critical points of the function. #tag_title# Step 8: Determine the range of the function #tag_content# The range of a function consists of all possible output values (y-values) resulting from the input values in the domain. Determining the range of the function u analytically may be challenging due to the complexity of the function. However, using numerical methods or graphing software, you can observe the behavior of the function and estimate the range. The range depends on the properties of the original sine and cosine functions, as well as the values of x.

Step-by-step solution

Understand the definitions of vers and covers

We are given the definitions of versed sine and coversed sine as follows: $$\text { vers } x = 1 - \cos x$$ $$\text { covers } x = 1 - \sin x$$ These definitions will be useful when simplifying the expression for u.

Examine the given function

The function u is defined as follows: $$u = x \sin 2x$$ Our goal is to analyze and simplify this function.

Use trigonometric identity to rewrite the function u

We can use the double angle formula for sine, which states that \(\sin 2x = 2 \sin x \cos x\). Using this identity, we can rewrite the function u as: $$u = 2x \sin x \cos x$$

Apply the given definitions to simplify the function

Recall the definitions of vers and covers: $$\text { vers } x = 1 - \cos x$$ $$\text { covers } x = 1 - \sin x$$ We can rewrite these equations as: $$\cos x = 1 - \text { vers } x$$ $$\sin x = 1 - \text { covers } x$$ Now, substitute these expressions into the function u: $$u = 2x(1 - \text { covers } x)(1 - \text { vers } x)$$

Analyze the simplified function

After simplifying the function u, we now have: $$u = 2x(1 - \text { covers } x)(1 - \text { vers } x)$$ This function represents the product of x, the coversed sine of x, and the versed sine of x. We can further analyze this function to understand its properties, such as its domain, range, and critical points.

Key concepts

Versed Sine

The versed sine, or versine, of an angle is a trigonometric function defined as \( \text{vers } x = 1 - \cos x \). It represents the difference between 1 and the cosine of an angle. This function helps calculate distances around circles and in surveying applications.

• Versed sine always has a value between 0 and 2 for angles between 0 and 360 degrees.
• It is related to the cosine function, highlighting how the concave or convex curvature of waves behaves.

In practical terms, let's think about the value of the cosine at key angles.

• For \( x = 0 \), \( \cos 0 = 1 \), so \( \text{vers } 0 = 1 - 1 = 0 \).
• For \( x = 90^\circ \), \( \cos 90^\circ = 0 \), so \( \text{vers } 90^\circ = 1 - 0 = 1 \).
• Finally, for \( x = 180^\circ \), \( \cos 180^\circ = -1 \), so \( \text{vers } 180^\circ = 1 - (-1) = 2 \).

By understanding these values, you'll have a clearer insight into how the versed sine function behaves and its significance.

Coversed Sine

The coversed sine, or coversine, relates to the sine function and complements the versed sine. It is given by \( \text{covers } x = 1 - \sin x \). The value of the coversed sine function measures how much is missing to reach a complete sine bracket, aiding in various geometric calculations.

• Coversed sine values range between 0 and 2, given the sine wave's range from -1 to 1.
• This function can be particularly useful in navigation and astronomical calculations where angles are key.

Looking at the coversed sine values for some key angles can facilitate understanding:

• For \( x = 0 \), \( \sin 0 = 0 \), so \( \text{covers } 0 = 1 - 0 = 1 \).
• For \( x = 90^\circ \), \( \sin 90^\circ = 1 \), so \( \text{covers } 90^\circ = 1 - 1 = 0 \).
• At \( x = 180^\circ \), \( \sin 180^\circ = 0 \), so \( \text{covers } 180^\circ = 1 - 0 = 1 \).

The coversed sine complements the variety of trigonometric tools used to address problems involving periodic and circular forms.

Double Angle Formula

The double angle formula is a powerful trigonometric identity that simplifies expressions like the sine of twice an angle. It states that \( \sin 2x = 2 \sin x \cos x \). This formula is particularly useful for manipulating trigonometric equations to examine more complex relationships or simplify calculations.

• It is derived from the sum of angles formula \( \sin(a+b) = \sin a \cos b + \cos a \sin b \).
• This relation helps break down calculations into simpler components, enhancing efficiency in solving trigonometric functions or equations.

Understanding how \( \sin 2x \) behaves at specific angles can offer significant insight:

• For \( x = 30^\circ \), \( \sin 60^\circ = \sin(2 \times 30^\circ) = \sqrt{3}/2 \), simplifying calculations without needing direct measurement.
• For \( x = 45^\circ \), \( \sin 90^\circ = 1 \), confirming the maximum amplitude reached by the sine wave.
• Finally, for \( x = 60^\circ \), \( \sin 120^\circ = \sin(2 \times 60^\circ) = \sqrt{3}/2 \), providing consistent trigonometric behavior insights.

The double angle formula is indispensable in mathematics and its applications, providing efficiency and accuracy in trigonometric computations.