Problem 21 - Algebra 2 and Trigonometry

Question

If \(\tan \theta\) is undefined, find all possible values of: \(\mathbf{a} . \cos \theta \quad \mathbf{b} . \sin \theta \quad \mathbf{c} . \cot \theta\)

Step-by-Step Solution

Answer

\( \cos \theta = 0 \); \( \sin \theta = (-1)^k \); \( \cot \theta \) undefined.

Step 1: Understanding Undefined Tangent

The function \( \tan \theta \) is undefined when the cosine of the angle \( \theta \) is zero because \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This occurs when \( \theta \) is an odd multiple of \( \frac{\pi}{2} \). Thus, \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

Step 2: Evaluating Cosine

For the angle \( \theta = \frac{\pi}{2} + k\pi \), cosine is zero. Therefore, \( \cos \theta = 0 \).

Step 3: Evaluating Sine

The sine function is \( 1 \) or \( -1 \) when \( \theta = \frac{\pi}{2} + k\pi \). Specifically, \( \sin \theta = (-1)^k \). This alternation is due to the periodic nature of the sine function with a period of \( 2\pi \).

Step 4: Evaluating Cotangent

Remember that \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). Since \( \cos \theta = 0 \), \( \cot \theta = 0 \). However, since \( \tan \theta \) is undefined entirely in these cases (division by zero), we consider \( \cot \theta \) as undefined as well.

Key Concepts

Trigonometric Identities

Trigonometric identities are the foundational building blocks for understanding relationships between trigonometric functions such as sine, cosine, and tangent. One of the essential identities you should know is that the tangent of an angle, represented as \( \tan \theta \), is the ratio of the sine and cosine functions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This identity helps clarify why \( \tan \theta \) becomes undefined; specifically, when \( \cos \theta = 0 \). When the cosine of an angle is zero, you end up dividing by zero, which is undefined in mathematics.
Understanding this identity is useful when solving exercises involving undefined tangent functions because it highlights why the tangent function can be problematic at certain angles. Next time, when you see an undefined tangent, think about its equation in terms of sine and cosine to unravel the mystery of why it behaves that way.

Undefined Trigonometric Functions

Undefined trigonometric functions occur when their values become impossible to calculate due to division by zero. The case of \( \tan \theta \) illustrates this beautifully, as it is undefined when \( \cos \theta = 0 \).
In these situations, knowing the specific angles involved helps. For tangent to be undefined, \( \theta \) must be an odd multiple of \( \frac{\pi}{2} \).

• Examples include \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \)

When you encounter undefined trigonometric functions, identify the angle first. Recognize that such angles lead to the numerator or denominator equalling zero in the identity involved, causing undefined behavior. It’s like trying to divide a total absence; it just doesn’t make sense in a tangible way. But this concept is central to understanding trigonometry's more complex behaviors.

Periodic Functions

Trigonometric functions, like sine and cosine, are periodic. This means they repeat their values in a consistent pattern over a regular interval. Specifically, the sine function has a period of \( 2\pi \). This property means that \( \sin \theta = \sin(\theta + 2k\pi) \) for any integer \( k \).
When dealing with undefined tangents, this periodicity is crucial. Notice how the sine function flips between \( 1 \) and \(-1\) as \( \theta \) behaves according to \( \theta = \frac{\pi}{2} + k\pi \). This alternation is because the effect of adding \( \pi \) shifts the angle by half a period of sine, thus reflecting its periodic nature.
With periodic functions, cycles repeat predictably. They inform a host of trigonometric calculations, especially when determining angle values over different periods. Embrace this repetition, as it simplifies tackling diverse trigonometric problems by showing how angles share specific properties cyclically.