Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Tell whether each statement is true or false. If false, tell why. The tangent and secant functions are undefined for the same values.

Step-by-Step Solution

Answer

True, tangent and secant are undefined for the same values.

Step 1: Understanding Tangent Function

The tangent function, denoted as \( \tan(x) \), is defined as the ratio of the sine function to the cosine function: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). The tangent function is undefined wherever the cosine function is zero.

Step 2: When is Cosine Zero?

The cosine function, \( \cos(x) \), is zero at odd multiples of \( \frac{\pi}{2} \). This means that \( \cos(x) = 0 \) for \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. Therefore, the tangent function is undefined for these values.

Step 3: Understanding Secant Function

The secant function, denoted as \( \sec(x) \), is the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \). The secant function is undefined wherever the cosine function is zero.

Step 4: Comparing Undefined Points

Both the tangent and secant functions are undefined wherever \( \cos(x) = 0 \). As derived earlier, these are the points \( x = \frac{\pi}{2} + n\pi \). Thus, both functions are indeed undefined for the same values.

Key Concepts

Tangent Function

The tangent function is a cornerstone in trigonometry and is frequently used to solve various geometric problems. It's defined as the ratio of the sine function to the cosine function: \[\tan(x) = \frac{\sin(x)}{\cos(x)}\] Understanding this relationship is crucial for mastering trigonometry. The tangent function reveals a unique property: it is undefined when the denominator, or cosine, is zero. At these points, the value of the tangent function approaches infinity, making it impossible to assign a finite number to it.

• This occurs at angles such as \( \frac{\pi}{2}, \frac{3\pi}{2} \), and so on.
• These angles are known as vertical asymptotes for the tangent graph because the tangent line shoots off to infinity.

When dealing with right triangles, the tangent of an angle can be described as the ratio of the opposite side to the adjacent side. This makes the tangent function invaluable in various applications such as architecture and physics.

Secant Function

The secant function is another significant trigonometric function, derived as the reciprocal of the cosine function. Mathematically, this is expressed as:\[\sec(x) = \frac{1}{\cos(x)}\]When the cosine of an angle is zero, taking its reciprocal becomes problematic since division by zero is undefined. Therefore, the secant function is also undefined in these instances.

• Just like the tangent function, the secant is undefined at angles such as \( x = \frac{\pi}{2} + n\pi \).
• These points, where secant becomes undefined, correspond to the vertical asymptotes of its graph.

The secant function, although less commonly used than the sine or cosine function, plays a critical role in calculus and complex analysis, particularly in solving integrals that cannot be expressed in simpler trigonometric terms.

Undefined Values

An undefined value in trigonometry typically arises when there is a division by zero or when a function approaches infinity. For the tangent and secant functions, undefined values occur precisely where the cosine function is zero.

• The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \).
• Both tangent and secant functions cannot have defined values at these points.

These undefined values are crucial in mathematics as they introduce the concept of limits and asymptotes. They help explain the behavior of functions near points of discontinuity, which often form the basis for advanced mathematical theorems and real-world modeling. Understanding where and why values become undefined aids in graphing and solving trigonometric equations, ensuring accurate results in calculations.

Cosine Function

The cosine function is fundamental in trigonometry and appears across numerous scientific and engineering disciplines. It defines how the projection of a rotating point onto the x-axis changes with angle. With the mathematical expression:\[\cos(x) = \text{adjacent side of the angle xxx quotient } \\text{in a right triangle over the hypotenuse.}\]Unlike the tangent and secant functions, the cosine function is defined for all real numbers, meaning it never becomes infinite. However, it does equal zero at specific points:

• These points are crucial for understanding when other functions, like tangent and secant, become undefined.
• The cosine function equals zero at points \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

The cosine curve is cyclic with a period of \( 2\pi \), producing a recognizable wave that oscillates between -1 and 1. This oscillation is key for understanding wave phenomena in physics and engineering, such as sound waves and electromagnetic waves. Its continuous and predictable nature makes it a reliable function for modeling periodic behavior.