Problem 29 - Algebra 2 and Trigonometry

Question

When tan \(\theta\) is undefined, find a value of: \(\quad\) a. \(\sin \theta \quad \mathbf{b} \cdot \cos \theta \quad \mathbf{c} . \theta\)

Step-by-Step Solution

Answer

\(\sin \theta = \pm 1\), \(\cos \theta = 0\), \(\theta = (2k+1)\frac{\pi}{2}\), where \(k\) is an integer.

Step 1: Understanding when tan(θ) is undefined

The tangent function, \(\tan \theta\), is undefined where the cosine function, \(\cos \theta\), is zero. This is because \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and division by zero is undefined. Cosine is zero at angles \((2k+1)\frac{\pi}{2}\) where \(k\) is any integer.

Step 2: Finding θ when tan(θ) is undefined

Given \(\cos \theta = 0\), \(\theta\) must be of the form \((2k+1)\frac{\pi}{2}\). The simplest case is \(\theta = \frac{\pi}{2}\), but other possibilities include \(\frac{3\pi}{2}, \frac{5\pi}{2}\), etc.

Step 3: Calculating sin(θ)

Now, substitute \(\theta = \frac{\pi}{2}\) into the sine function. We know \(\sin \frac{\pi}{2} = 1\). Therefore, when \(\tan \theta\) is undefined, \(\sin \theta = 1\).

Step 4: Calculating cos(θ)

Use \(\theta = \frac{\pi}{2}\) again. We have \(\cos \frac{\pi}{2} = 0\). This was expected since \(\tan \theta\) being undefined depends on \(\cos \theta = 0\).

Step 5: Identifying θ for other possibilities

For other values like \(\theta = \frac{3\pi}{2}\), you can check: \(\sin \frac{3\pi}{2} = -1\) and \(\cos \frac{3\pi}{2} = 0\). Similarly, \(\theta = \frac{5\pi}{2}\) gives \(\sin \theta = 1\) and \(\cos \theta = 0\).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is an essential part of trigonometry. It connects the sine and cosine functions in one simple relationship: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This formula tells us that the tangent is the quotient of sine and cosine for a given angle \( \theta \). This relationship means that tangent is directly dependent on the values of both sine and cosine.

Understanding the behavior of tangent involves recognizing its periodicity and the instances where it becomes undefined. The tangent completes a full cycle at \( \pi \), repeating its pattern indefinitely beyond that. It has an infinite range, unlike sine and cosine, which both lie between -1 and 1.

The situations where \( \tan \theta \) becomes undefined arise when \( \cos \theta = 0 \). Since division by zero is undefined in mathematics, wherever the cosine values reach zero, the tangent function cannot be computed. This typically occurs at specific angles, making it vital to understand both sine and cosine to confidently work with the tangent function.

Undefined Expressions

In mathematics, especially when dealing with functions and calculations, the term "undefined" is encountered when an operation cannot be performed due to mathematical rules. In the context of trigonometric functions, an expression becomes undefined when it involves a division by zero.

For the tangent function, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), the expression is undefined where \( \cos \theta \) is zero. This happens because division by zero is prohibited in mathematics—it disrupts the calculation and leads to indefinable results.

Angles at which \( \cos \theta = 0 \) are of the form \( (2k+1)\frac{\pi}{2} \), where \( k \) is any integer. At these angles, the value of cosine hits zero, thus making tangent undefined. Identifying such angles is crucial for understanding not just the tangent function, but the interplay between all trigonometric functions.

Sine and Cosine Values

The sine and cosine functions are fundamental in trigonometry, each describing waves that oscillate between -1 and 1.

The sine function, \( \sin \theta \), measures the vertical component of an angle's rotation on the unit circle, achieving its maximum at \( \theta = \frac{\pi}{2} \), where \( \sin \frac{\pi}{2} = 1 \). Its periodic nature repeats every \( 2\pi \), creating a consistent wave pattern.

The cosine function, \( \cos \theta \), represents the horizontal component of that angle, hitting zero at \( (2k+1)\frac{\pi}{2} \), which makes it crucial in determining when the tangent function enters into an undefined state.

By understanding sine and cosine's complementary relationship, you can gauge their impact on other functions like tangent. When \( \cos \theta = 0 \), knowing the sine function allows for complete insight into the behavior of the tangent function—and vice versa—despite cosine's zero status impacting tangent's definition.