Problem 49 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\tan \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi$$

Step-by-Step Solution

Answer

\( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \)

Step 1: Understanding the Tangent Function

The tangent of an angle \( \theta \) on the unit circle is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The tangent function is undefined when the denominator, \( \cos \theta \), is zero.

Step 2: Identify the Undefined Condition

For \( \tan \theta \) to be undefined, \( \cos \theta = 0 \). On the unit circle, the cosine function equals zero at certain standard angles.

Step 3: Locate Angles with Zero Cosine

The angles where \( \cos \theta = 0 \) on the unit circle are at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). These are the points where the x-coordinate (cosine value) of the unit circle is zero, causing the tangent function to be undefined.

Step 4: Confirm the Interval

Check that both \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) are within the interval \( 0 \leq \theta \leq 2\pi \). Both are within the given range, thus valid solutions.

Key Concepts

Tangent Function

The tangent function, often represented as \( \tan \theta \), is a fundamental concept in trigonometry. It is defined as the ratio of the sine and cosine functions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) This means that the value of the tangent function depends on both the sine and cosine values of the angle \( \theta \).

• Sine Function: The vertical coordinate on the unit circle.
• Cosine Function: The horizontal coordinate on the unit circle.

The tangent function is key in many trigonometric calculations and appears in various mathematical settings. It represents how steep a line is in relation to the horizontal axis when the line corresponds to the angle \( \theta \). Understanding the basic relationship between sine, cosine, and tangent is crucial for interpreting the behavior of angles on the unit circle.

Undefined Tangent

The tangent function becomes undefined whenever the cosine function equals zero, since division by zero is not possible. On the unit circle, this happens at angles where the x-coordinate, the cosine value, is zero.

• When \( \cos \theta = 0 \), \( \tan \theta \) is undefined.
• For an undefined tangent, look for angles on the unit circle where this occurs.

These specific angles are \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). At these points, the cosine value is zero, creating a situation where the sine-to-cosine ratio is undefined. Understanding when and why \( \tan \theta \) becomes undefined is important as it helps identify gaps in continuity on graphs of the tangent function and interprets angles where the function cannot provide a meaningful value.

Cosine Function

The cosine function, denoted \( \cos \theta \), is an important trigonometric function playing a pivotal role in the unit circle.

• Definition: The x-coordinate of a point on the unit circle corresponding to an angle \( \theta \).
• Properties: Periodic with a cycle of \( 2\pi \), values range between -1 and 1.

Cosine is a function that determines how far along the horizontal axis a point lies as one moves around the unit circle. When studying the unit circle, seeing when \( \cos \theta = 0 \) is essential for identifying angles where the tangent function is undefined. At \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), the cosine value is exactly zero, directly affecting the tangent's defined status. Allowing recognition of these zero-crossing points can simplify solving trigonometric equations and identify key characteristics of wave behavior.

Angle Measurement

Angles are measured in radians on the unit circle, a convenient and natural measure in trigonometry. A radian angle considers the length of the arc created by the angle on the circle's circumference.

• Full Circle: Equals \(2\pi\) radians, equivalent to 360 degrees.
• Key Angles: \( \frac{\pi}{2} = 90^\circ \) and \( \frac{3\pi}{2} = 270^\circ \).

When working within the interval \(0 \leq \theta \leq 2\pi\), we examine angles within a complete revolution around the circle. Calculating angles in radians simplifies mathematical operations and connects seamlessly with concepts such as arc length and circular motion. Recognizing radians and how they relate to degrees provides a strong foundation for exploring trigonometric functions and identifying conditions like undefined tangent values at key non-standard angles.