Question

Multiple Choice Which of the following functions is even? (a) cosine (b) sine (c) tangent (d) cosecant

Short answer

cosine

Step-by-step solution

Define Even and Odd Functions

Even functions satisfy the property that for all values of x, f(x) = f(-x). Odd functions satisfy the property f(x) = -f(-x).

Analyze Cosine Function

Consider \(f(x) = \cos(x)\). Calculate \(\cos(-x)\). Since \(\cos(-x) = \cos(x)\), the cosine function is even.

Analyze Sine Function

Consider \(f(x) = \sin(x)\). Calculate \(\sin(-x)\). Since \(\sin(-x) = -\sin(x)\), the sine function is odd.

Analyze Tangent Function

Consider \(f(x) = \tan(x)\). Calculate \(\tan(-x)\). Since \(\tan(-x) = -\tan(x)\), the tangent function is odd.

Analyze Cosecant Function

Consider \(f(x) = \csc(x)\). Calculate \(\csc(-x) = \frac{1}{\sin(-x)}\). Since \(\csc(-x) = -\csc(x)\), the cosecant function is odd.

Conclusion

From the analysis above, the cosine function \(\cos(x)\) is the only even function among the given options.

Key concepts

cosine function

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function. It's important to recognize one key feature about the cosine function: it is an even function. This means that the function satisfies the property \( \cos(-x) = \cos(x) \). In simple terms, if you reflect the function across the y-axis, it will be symmetrical. This characteristic makes it unique among the basic trigonometric functions.

One common way to visualize this is to think of the graph of the cosine function. If you graph \( \cos(x)\) and then graph \( \cos(-x)\), you will notice that they perfectly overlap.

Additionally, this function plays a crucial role in the construction of the unit circle and helps in defining other trigonometric functions. Understanding the properties of the cosine function is essential for mastering trigonometry.

sine function

The sine function, denoted as \( \sin(x) \), is another fundamental trigonometric function. Unlike the cosine function, the sine function is an odd function. This means it satisfies the property \( \sin(-x) = -\sin(x) \).

In practical terms, if you reflect the sine function across the y-axis, and then invert it (flip it upside down), it will match the original function. This symmetry is crucial in many applications, especially in engineering and physics.

Visualizing the graph of the sine function can help understand this property. When graphed, \( \sin(x) \) and \( -\sin(-x) \) will overlap, showing this unique symmetry. The sine function is central to wave mechanics and signal processing, making it essential to understand its properties thoroughly.

tangent function

The tangent function, denoted as \( \tan(x) \), arises from the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

Like the sine function, the tangent function is also an odd function, satisfying the property \( \tan(-x) = -\tan(x) \). This implies that reflecting it across the y-axis and then inverting it will yield the original function.

The graph of the tangent function displays this property. The function has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \), where it approaches infinity. These asymptotes emphasize the odd symmetry, as the function repeats its pattern every \( \pi \) units along the x-axis.

Understanding the behavior and properties of the tangent function is critical for solving trigonometric equations and analyzing periodic phenomena.

cosecant function

The cosecant function, denoted as \( \csc(x) \), is the reciprocal of the sine function: \( \csc(x) = \frac{1}{\sin(x)} \).

Similar to the sine function, the cosecant function is an odd function, adhering to the property \( \csc(-x) = -\csc(x) \). Reflecting and inverting it will result in the original function, highlighting its every-other symmetry across the y-axis.

The graph of the cosecant function features undefined points where the sine function is zero (at multiples of \( \pi \)), resulting in vertical asymptotes at these points. This graph helps to visualize the odd nature of the function.

Grasping the properties of the cosecant function is important for understanding more complex trigonometric identities and solving equations involving trigonometric functions.