Question

MAKING AN ARGUMENT Your friend states that it is not possible to write a cosecant function that has the same graph as \(y=\sec x\). Is your friend correct? Explain your reasoning.

Short answer

Yes, your friend is correct. It is impossible to write a cosecant function that has a same graph as \(y=\sec(x)\), because the cosecant and secant functions have different intervals of undefined points and the shapes of their waves are distinct.

Step-by-step solution

Understanding the wave nature and symmetry of the functions

The secant function \(y=\sec(x)\), has vertical asymptotes (undefined points) at \(x = (2n+1)\frac{\pi}{2}\), where n is any integer, and \(y=\csc(x)\), the cosecant function, has vertical asymptotes at \(x = n\pi\), where n is any integer. Therefore, the secant function is undefined for an odd multiple of \(\pi/2\), whereas the cosecant function is undefined for any multiple of \(\pi\). Hence, the interval between two consecutive undefined points differs in each of these functions, causing different symmetry in the graphs.

Comparing the graphs of sec and csc functions

Looking at the graphs of the two functions, the secant function creates a wave that opens upward and downward, whereas the cosecant function creates a wave solely opening downwards or upwards. Their separate specific undefined points and the direction of open ends make the graphs distinct from each other and they cannot be similar.

Making a conclusion

Based on the interval of undefined points for both functions and the different shape of waves produced by them, it is impossible to write a cosecant function that has the same graph as \(y=\sec(x)\). Thus, your friend is correct.

Key concepts

Trigonometric Functions

Understanding trigonometric functions is essential for delving into the world of trigonometry. These functions, including sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent, are used to relate the angles of a triangle to the lengths of its sides in a right-angled triangle.

The cosecant function, denoted as \(y=\csc(x)\) or \(\csc x\), is the reciprocal of the sine function, meaning \(\csc x = \frac{1}{\sin x}\) when \(\sin x \eq 0\). Similarly, the secant function, \(y=\sec(x)\) or \(\sec x\), is the reciprocal of the cosine function, or \(\sec x = \frac{1}{\cos x}\) when \(\cos x \eq 0\).

When graphing these functions, it is important to note that because they are reciprocals of sine and cosine, their values will be undefined where sine and cosine are zero. This leads to the occurrence of vertical asymptotes in their respective graphs.

Graphical Symmetries

Graphical symmetries in trigonometric functions can give us clues about their behavior and properties. Symmetry in mathematics implies that one half of the graph is a mirror image of the other half, or that the graph repeats itself after a certain interval — a feature known as periodicity.

The cosecant and secant functions exhibit different types of symmetries. The secant function, \(y=\sec(x)\), is symmetrical about the y-axis for even functions and origin for odd functions, possessing a period of \(2\pi\). The graph of the secant function is not a continuous curve but consists of repeating patterns of upward and downward waves, separated by vertical asymptotes.

The cosecant function, \(y=\csc(x)\), similarly has symmetry and periodicity with a discontinuous graph that consists of alternating segments that curve downwards or upwards, also separated by vertical asymptotes. These functions' symmetrical properties ensure the predictability and repetitive nature of their graphs across the coordinate plane.

Vertical Asymptotes

Vertical asymptotes are lines that a function's graph approaches but never touches or crosses. They correspond to the x-values where the function is undefined and indicate where the graph shoots up to infinity or drops down to negative infinity.

Within the context of the secant and cosecant functions, vertical asymptotes occur at the zeroes of their respective reciprocal functions, sine and cosine. For \(y=\sec(x)\), vertical asymptotes are found at \(x = (2n+1)\frac{\pi}{2}\) for every integer value of n, whereas for \(y=\csc(x)\), the asymptotes occur at \(x = n\pi\) for every integer n.

These asymptotes play a crucial role in determining the distinctive wave patterns you see on the graph. A crucial point to understand is that the presence and location of vertical asymptotes help define the unique periodic nature of these functions and are a key element in their graphical representation.

Trigonometric Identities

Trigonometric identities are equations that express one trigonometric function in terms of others. These identities demonstrate the relationships between the functions and enable us to simplify complex trigonometric expressions and solve trigonometric equations.

Some of the fundamental identities include the reciprocal identities, like \(\csc x = \frac{1}{\sin x}\) and \(\sec x = \frac{1}{\cos x}\), and the Pythagorean identities, such as \(\sin^2 x + \cos^2 x = 1\) and its derivatives \(\tan^2 x + 1 = \sec^2 x\) and \(1 + \cot^2 x = \csc^2 x\).

Understanding these identities is crucial when analyzing trigonometric functions, as they help us transition between different functions and can simplify the solving of trigonometric equations. By using an identity, we may find alternative expressions for a function that are more convenient for a particular purpose, which is a powerful tool in trigonometry.