Problem 79 - 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. $$\tan \left(2 n \cdot 90^{\circ}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Identify the expression

The given expression is \(\tan(2n \cdot 90^{\circ})\). This suggests that the tangent is evaluated at an angle which is a multiple of \(90^{\circ}\).

Step 2: Simplify and analyze the angle

We know the tangent function, \(\tan(\theta)\), is defined except for angles where \(\theta\) leads to vertical asymptotes, specifically odd multiples of \(90^{\circ}\), where it's undefined. Let's analyze:\(2n \cdot 90^{\circ} = 180^{\circ} \cdot n\).

Step 3: Determine whether tangent is defined

Since \(180^{\circ} \cdot n\) is an integer multiple of \(180^{\circ}\), the angle is \(0^{\circ}, \pm180^{\circ}, \pm360^{\circ}\), etc. At these angles, tangent is defined and equals 0 because tangent of any integer multiple of \(180^{\circ}\) is zero.

Key Concepts

Tangent Function

The tangent function, denoted by \( \tan(\theta) \), is a crucial trigonometric function that relates the angle \( \theta \) to the ratio of the opposite side to the adjacent side in a right-angled triangle. The function has a periodicity of \(180^{\circ}\), meaning \( \tan(\theta + 180^{\circ}) = \tan(\theta) \). This makes the function repeat its values every \(180^{\circ}\). As a result:

• It is essential to know that the tangent function is periodic and its graph repeats every \(180^{\circ}\).
• The tangent function can be undefined at certain points where it has vertical asymptotes.
• Unlike sine and cosine, which oscillate between -1 and 1, tangent can take any real value, which makes its graph distinct.

To apply this in the exercise, when calculating \( \tan(2n \cdot 90^{\circ}) \), the periodicity helps us simplify the problem, knowing the outcomes of the function at key multiples of \(90^{\circ}\) and \(180^{\circ}\).

Angle Multiples

Understanding angle multiples, especially in trigonometric functions, helps simplify complex expressions. When evaluating \( \tan(2n \cdot 90^{\circ}) \), it translates into finding tangent values at integer multiples of \(180^{\circ}\), such as \(0^{\circ}, \pm180^{\circ}, \pm360^{\circ} \), etc.

• Multiples of \(180^{\circ}\) directly influence the value of tangent, yielding results that are straightforward to anticipate.
• For every integer \(n\), \(2n \times 90^{\circ} = 180^{\circ} \cdot n\) perfects exactly these values at which tangent is defined and equals 0.
• This consistency provides a way to predict results without needing calculators for every specific case, leveraging the symmetry of the tangent function along its plotted curve.

These angles, multiples of \(180^{\circ}\), ensure the function evaluates safely—meaning not leading to undefined points or asymptotes in tangent function plots.

Function Undefined

Trigonometric functions can be undefined at specific points; the tangent function in particular is undefined at odd multiples of \(90^{\circ}\) like \(90^{\circ}, 270^{\circ}, 450^{\circ}\), etc. These points on the unit circle lead to vertical asymptotes.

• When \( \tan(\theta) \) approaches these odd multiples of \(90^{\circ}\), the tangent function stretches indefinitely both positively and negatively, which is why it's deemed undefined.
• For the exercise given, calculating \( \tan(2n \cdot 90^{\circ}) \), the expression simplifies to multiples of \(180^{\circ}\), avoiding the critical points where tangent is undefined.
• This calculation method allows more focus on where the tangent is defined and actionable (i.e., at \(180^{\circ}, \pm360^{\circ}, \pm540^{\circ}\), where the outcome is zero angularly), preventing misinterpretation from unsolvable positions.

Understanding where functions are undefined assists in resolving more complex equations or signals potential problem areas in calculations requiring preemptive adjustments.