Question

. Which of the following represent the same graph? Check your result analytically using trigonometric identities. (a) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (b) \(y=\cos \left(x+\frac{\pi}{2}\right)\) (c) \(y=-\sin (x+\pi)\) (d) \(y=\cos (x-\pi)\)

Short answer

Options (a) and (c) represent the same graph, but different from (b) and (d).

Step-by-step solution

Compare the function in option (a) with basic identities

The function is given by \(y = \sin \left(x+\frac{\pi}{2}\right)\). Using the identity \(\sin(x + \frac{\pi}{2}) = \cos(x)\), we can rewrite it as \(y = \cos(x)\).

Compare the function in option (b) with basic identities

The function is given by \(y = \cos \left(x+\frac{\pi}{2}\right)\). Using the identity \(\cos(x + \frac{\pi}{2}) = -\sin(x)\), we can rewrite it as \(y = -\sin(x)\).

Compare the function in option (c) with basic identities

The function is given by \(y = -\sin(x + \pi)\). Using the identity \(\sin(x + \pi) = -\sin(x)\), we can rewrite it as \(y = -(-\sin(x)) = \sin(x)\).

Compare the function in option (d) with basic identities

The function is given by \(y = \cos(x - \pi)\). Using the identity \(\cos(x - \pi) = -\cos(x)\), we can rewrite it as \(y = -\cos(x)\).

Compare the rewritten functions

After using trigonometric identities, we have: - (a) \(y = \cos(x)\) - (b) \(y = -\sin(x)\) - (c) \(y = \sin(x)\) - (d) \(y = -\cos(x)\) The functions in options (b) and (c) represent different graphs, and (a) and (d) represent different graphs too.

Key concepts

Graphs of Trigonometric Functions

Trigonometric functions like sine and cosine have distinct periodic graphs that repeat every \(2\pi\) radians. Understanding the graphs of these functions is crucial for recognizing patterns and transformations. For the sine function, the graph starts at zero, reaches a maximum at \(\pi/2\), returns to zero at \(\pi\), and has a minimum at \(3\pi/2\), before completing the cycle at \(2\pi\). Contrastingly, the cosine function begins at its maximum, then follows a similar pattern as the sine but shifted to the left by \(\pi/2\).
These graphs help visualize how trigonometric functions react to changes in their arguments. Observing transformations like vertical shifts, stretches, or horizontal shifts (phase shifts) becomes evident when viewing their respective curves on a graph.
Analyzing trigonometric graphs provides insight into their functionality and use in various applications, including physics and engineering. Knowing how to interpret these graphs makes identifying key properties of functions more intuitive.

Phase Shift in Trigonometry

Phase shift refers to the horizontal movement of a trigonometric graph along the \(x\)-axis. This concept is essential when comparing and rewriting trigonometric functions. It's determined by the change in the function's argument, typically expressed as a constant added or subtracted to \(x\). For example, in the function \(y = \sin(x + \frac{\pi}{2})\), the \(\frac{\pi}{2}\) term indicates a phase shift.
A positive phase shift, as seen in \(\sin(x + \frac{\pi}{2})\), moves the graph to the left. Conversely, a negative phase shift, such as in \(\cos(x - \pi)\), shifts the graph to the right. These shifts adjust the starting point of the function's usual cycle on the graph.
Understanding phase shifts allows you to rewrite trigonometric functions in different forms, making them easier to compare and analyze against each other. Applying identities, such as how \(\sin(x + \frac{\pi}{2})\) becomes \(\cos(x)\), showcases how phase shifts affect function behavior and appearance on the graph.

Comparing Trigonometric Functions

Comparing trigonometric functions, particularly when involving phase shifts, calls for the application of trigonometric identities. These identities simplify functions into recognizable forms. Through identities like \(\sin(x + \frac{\pi}{2}) = \cos(x)\) or \(\cos(x + \frac{\pi}{2}) = -\sin(x)\), functions can be directly compared. This method highlights equivalences or differences among functions.
Effective comparison also uses the graphical approach, observing if two functions plot the same or different curves. As in our original problem, rewriting different expressions of sine and cosine established which ones coincide graphically. For example, \(y = \cos(x)\) (from \(y = \sin(x + \frac{\pi}{2})\)) doesn't match \(y = \sin(x)\), while \(y = -\cos(x)\) (from \(y = \cos(x - \pi)\)) produces a distinct graph.
By understanding these transformations and their resulting graphs, you gain the ability to predict and describe function behaviors simply by analyzing their equations and using identities.