Problem 80 - A Graphical Approach to Precalculus with Limits

Question

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\tan \left[(2 n+1) \cdot 90^{\circ}\right]$$

Step-by-Step Solution

Answer

The expression is undefined.

Step 1: Understand the given expression

The expression given is \( \tan \left[(2n+1) \cdot 90^{\circ}\right] \). This expression involves the tangent function evaluated at an angle, which is an odd integer multiple of \( 90^{\circ} \).

Step 2: Analyze the pattern of tangent at these angles

Recall that the tangent function, \( \tan(\theta) \), has a known pattern: \( \tan(90^{\circ} + k \cdot 180^{\circ}) = \text{undefined} \) for integer \( k \). The expression \((2n+1) \cdot 90^{\circ}\) fits the form \(90^{\circ} \cdot (2n+1)\) which results in an angle that matches this pattern.

Step 3: Determine the parity of the angle

For the angle \((2n+1) \cdot 90^{\circ}\), it is evident that the angle is odd multiples of \( 90^{\circ} \). Thus, it represents angles like \( 90^{\circ}, 270^{\circ}, 450^{\circ}, \) etc.

Step 4: Conclude tangential properties at these angles

For these specific angles (\( 90^{\circ}, 270^{\circ} \) etc.), the tangent function is undefined because it involves division by zero in the tangent function, which makes \( \tan \theta \) undefined at odd multiples of \( 90^{\circ} \).

Key Concepts

Tangent Function

The tangent function, represented as \( \tan(\theta) \), is a fundamental trigonometric function that arises from a right-angled triangle. In the triangle, the tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the adjacent side. It can also be described as the sine of the angle divided by the cosine of the angle: \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\] Unlike sine and cosine functions, which oscillate between -1 and 1, the tangent function can take on any real value. The function is periodic with a period of \( 180^{\circ} \) (or \( \pi \) radians), which means it repeats its values every \( 180^{\circ} \) rotation. Understanding its behavior at specific angles can reveal much about the nature of its graph and undefined values.

Undefined Values

The tangent function has specific angles where it becomes undefined. This occurs because division by zero in the computation of tangent makes it impossible to define a numeric value. These undefined values happen when the angle \( \theta \) results in a cosine of zero, since the form \( \frac{\sin(\theta)}{\cos(\theta)} \) implies a division by zero.

• Tangent is undefined at angles where \( \cos(\theta) = 0 \), corresponding to \( \theta = 90^{\circ}, 270^{\circ}, 450^{\circ}, \ldots \)
• These are represented as odd multiples of \( 90^{\circ} \): \((2n+1) \times 90^{\circ} \), where \( n \) is an integer.

Because it's undefined at these angles, the tangent function's graph has vertical asymptotes that illustrate these discontinuities. These asymptotes are essential for visualizing why the tangent does not exist at certain points.

Odd Multiples

An odd multiple of an angle like \( 90^{\circ} \) is formed by multiplying the angle by an odd integer expression, such as \( (2n+1) \times 90^{\circ} \). This results in angles like \( 90^{\circ}, 270^{\circ}, 450^{\circ}, \) and so forth.

• Odd multiples of \( 90^{\circ} \) are crucial as they signify points where the tangent function is undefined.
• When addressing expressions like \( \tan\left[(2n+1) \cdot 90^{\circ}\right] \), it indicates a pattern where tangible numeric values for tangent don’t exist.

These odd multiples are essential in studying the behavior of trigonometric functions, particularly how equations behave at key angular points. Identifying these points allows for a deeper understanding of the periodic nature and characteristics of tangent, especially where it does not produce results.