Problem 101 - Algebra and Trigonometry

Question

Graph each equation in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right] b y\) \([-3,3,1]\) viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent. $$ y=\csc x-\cot x $$

Step-by-Step Solution

Answer

The equation \(y=\csc x-\cot x\) has the same graph as \(y=\frac{1-\cos x}{\sin x}\), and this is verified by their graphs coinciding when plotted within the same viewing rectangle.

Step 1: Graph the functions

Starting off, by using a graphing tool, the equation, \(y=\csc x-\cot x\), is plotted in the viewing rectangle \([-2 \pi, 2 \pi,\frac{\pi}{2}]\) by \([-3,3,1]\). This will give the overall idea on how the equation looks like when plotted.

Step 2: Describe the graph using another equation

To represent the same graph using another equation, we need to take advantage of trigonometric identities. Using the identity that \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\), the given equation can be rewritten as \(y = \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \frac{1 - \cos x}{\sin x}\). This is a different equation depicting the same graph.

Step 3: Verifying the Equivalent Equations

To verify that both the equations, \(y = \csc x - \cot x\) and \(y = \frac{1 - \cos x}{\sin x}\) are equivalent, they are plotted separately on the graph within the given viewing rectangle. If the graphs coincide and give us the same pattern, we can safely conclude that both equations are equivalent. Now comparing graphs demonstrates they are equivalent.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying expressions and solving trigonometry problems. These identities are equations that involve the trigonometric ratios like sine, cosine, tangent, etc., and are true for all values of the variables where both sides of the identity are defined.

For example, the Pythagorean identities are an integral set of identities: \(\sin^2 x + \cos^2 x = 1\), \(1 + \tan^2 x = \sec^2 x\), and \(1 + \cot^2 x = \csc^2 x\). These are used to convert from one form of trigonometric function to another, facilitating the process of graphing or solving equations.

In our exercise, identities like \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\) were employed to describe the graph of \(y=\csc x-\cot x\) using a different, yet equivalent, equation.

Cosecant Function

The cosecant function, denoted by \(\csc x\), is the reciprocal of the sine function. It's defined as \(\csc x = \frac{1}{\sin x}\), assuming that \(\sin xeq 0\).

The graph of the cosecant function is made up of arcs that go to infinity whenever the sine function is zero because division by zero is undefined. These points are called vertical asymptotes of the cosecant function and are crucial when graphing. It is important in trigonometry because it gives us a relationship between the angle and the ratio of the length of the hypotenuse to the opposite side in a right-angled triangle.

Cotangent Function

Similarly, the cotangent function, symbolized as \(\cot x\), is the reciprocal of the tangent function and is defined as \(\cot x = \frac{\cos x}{\sin x}\), given that \(\sin x\) is not zero.

The graph of \(\cot x\) has its own unique characteristics, such as undefined values at \(\sin x = 0\), which creates vertical asymptotes, similar to the cosecant function. These discontinuities should be mindfully marked when graphing the cotangent function. It arises often in mathematics, representing the ratio of the adjacent side to the opposite side in a right-angled triangle.

Equivalent Equations

Equivalent equations are distinct algebraic expressions that yield the same solution set or represent the same graph when plotted on a coordinate plane. They are typically used to illustrate different aspects of the same mathematical relationship or to simplify complex expressions for easier analysis.

In graphing trigonometric functions, we can use trigonometric identities to rewrite equations into equivalent forms that may be more straightforward to graph or analyze. In our exercise example, we transformed \(y=\csc x-\cot x\) into an equivalent equation \(y=\frac{1-\cos x}{\sin x}\) using identities, which helped us verify the equivalence by comparing the graphs.

Graphing Techniques in Trigonometry

Graphing is a vital technique in trigonometry, as it provides a visual representation of functions and their characteristics. There are several strategies to graph trigonometric functions effectively.

• Understanding the basic shape and period of the function.
• Knowing the key points, such as the zeros, maximums, and minimums.
• Being aware of any asymptotes and discontinuities.
• Applying transformations such as shifts, stretches, and reflections correctly.

In our graphing exercise, we used the properties of the cosecant and cotangent functions to plot \(y=\csc x-\cot x\) within a specified viewing rectangle. Recognizing the individual characteristics of each trigonometric function allowed us to accurately depict their combine influence and verify their equivalence visually.