Question

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

Short answer

In this exercise, the definitions of the versed sine and coversed sine functions were given, along with a formula involving tangent functions. Following a step-by-step process, we derived a formula involving both versed sine and coversed sine functions. The final formula is given by: $$\sin \theta - \sin \theta \cdot \text{vers} \phi + \sin \phi - \sin \phi \cdot \text{covers} \theta = 2 \sin \theta \sin \phi$$

Step-by-step solution

Understanding the versed sine and coversed sine functions

Given that the versed sine is defined as follows: $$\text { vers } x=1-\cos x$$ And the coversed sine is defined as: $$\text { covers } x=1-\sin x$$ These are simply the complements of the cosine and sine functions, respectively.

Expressing tangent functions in terms of sine and cosine functions

Given the formula: $$\tan \theta+\tan \phi=2 \tan \phi \tan \theta$$ Recall that the tangent function is defined as the sine function divided by the cosine function. Therefore, the given formula can be rewritten as: $$\frac{\sin \theta}{\cos \theta} + \frac{\sin \phi}{\cos \phi} = 2 \cdot \frac{\sin \theta \sin \phi}{\cos \theta \cos \phi}$$

Finding a common denominator and combining the fractions

In order to simplify the equation from Step 2, we need to find a common denominator for the fractions so they can be combined. The common denominator is \(\cos \theta \cos \phi\). So rewrite the equation as: $$\frac{\sin \theta \cos \phi + \sin \phi \cos \theta}{\cos \theta \cos \phi} = 2 \cdot \frac{\sin \theta \sin \phi}{\cos \theta \cos \phi}$$

Equate the numerators

Since the denominators on both sides of the equation are equal, we can equate the numerators: $$\sin \theta \cos \phi + \sin \phi \cos \theta = 2 \sin \theta \sin \phi$$

Substitute versed sine and coversed sine functions for cosine and sine

Now we will use the definitions of the versed sine and coversed sine functions to substitute cosine and sine functions back into the equation: $$\sin \theta (1 - \text{vers} \phi) + \sin \phi (1 - \text{covers} \theta) = 2 \sin \theta \sin \phi$$

Simplifying the equation

Distribute the sine functions on the left side: $$\sin \theta - \sin \theta \cdot \text{vers} \phi + \sin \phi - \sin \phi \cdot \text{covers} \theta = 2 \sin \theta \sin \phi$$ Now, we have the desired equation with the versed sine and coversed sine functions involved.

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental to the study of geometry and physics, bridging the relationship between angles and sides of triangles. At the core, the primary trigonometric functions are sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}), where each function has a specific mathematical definition related to a right-angled triangle.

For any angle \(\theta\), the sine function is defined as the ratio of the length of the side opposite \(\theta\) to the length of the hypotenuse, while the cosine is the length of the adjacent side over the hypotenuse. The tangent function is particularly interesting as it represents the ratio of the sine and cosine of the angle, or in other words, the ratio of the opposite side to the adjacent side.

\text{Sine} and \text{cosine} have values that range from -1 to 1, as they represent ratios of sides in a right-angled triangle. These functions are also periodic, repeating their values in regular intervals, which is a characteristic pivotal in the study of oscillations and waves. An extension of these concepts, the \text{versed sine} and \text{coversed sine}, are less commonly used but provide additional relationships within trigonometry, as shown in the homework exercise.

Tangent Function

The tangent function is an intrinsic aspect of trigonometric functions and provides a distinct relationship between sine and cosine. The formula for the tangent of an angle \(\theta\) is \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). What makes the tangent function so special is its behavior; unlike sine and cosine, the tangent function does not have a maximum or minimum value and is not bound between -1 and 1.

In practical applications, the tangent function illustrates the slope of a line relative to the x-axis, among other things. Additionally, when it comes to solving equations or inequalities involving tangent, it is often useful to rewrite it in terms of sine and cosine to utilize their properties or for the sake of integration or differentiation, as seen in the exercise where \(\tan\theta + \tan\phi\) is further explored.

Understanding how to manipulate the tangent function using algebra and trigonometric identities, such as in the original exercise, is crucial in solving more complex trigonometric problems.

Sine and Cosine Relationships

The relationship between sine (\text{sin}) and cosine (\text{cos}) functions is deep-rooted in the study of trigonometry. These functions are co-dependent, meaning that the knowledge of one can provide information about the other. One of the most fundamental identities involves the Pythagorean theorem, which states that \(\sin^{2}(x) + \cos^{2}(x) = 1\).

This relationship underpins almost all trigonometric concepts and is used to derive other important identities, such as those for tangent, secant, cosecant, cotangent, and, of course, the versed sine and coversed sine.

Moreover, there's a useful phase shift relationship between sine and cosine: \(\cos(x) = \sin(x + \frac{\pi}{2})\). Recognizing these relationships empowers students to transform and simplify trigonometric expressions, solve equations, and understand the behaviors of these functions, as they cyclically complement one another. As students engage with exercises like the one provided, their ability to see the interconnectedness of these functions grows, unlocking advanced methods to approach trigonometry problems.