Problem 2 - Algebra 2 and Trigonometry

Question

What is the value of \(\cos \theta\) when \(\tan \theta\) is undefined? Justify your answer.

Step-by-Step Solution

Answer

The value of \(\cos \theta\) is 0 when \(\tan \theta\) is undefined.

Step 1: Understanding Tangent Undefined

The tangent of an angle, \(\tan \theta\), is undefined when the denominator of its ratio is zero. Tangent is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, \(\tan \theta\) is undefined when \(\cos \theta = 0\).

Step 2: Determine When Cosine is Zero

The cosine function equals zero at specific points on the unit circle. \(\cos \theta = 0\) at \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. These angles lie along the vertical axis of the unit circle, where the x-coordinate of the point is zero.

Step 3: Analyze the Value of Cosine

When \(\cos \theta = 0\), the \(\theta\) being examined places the point on the unit circle directly on the y-axis. This ensures that the angle causes the cosine value to reach zero, confirming that this is the mathematical reason \(\tan \theta\) becomes undefined.

Key Concepts

Cosine Function

The cosine function plays a crucial role in trigonometry, especially when analyzing periodic functions on the unit circle. The cosine function, denoted as \( \cos \theta \), essentially measures the horizontal distance or x-coordinate of a point on the unit circle.

It's important to note:

• The value of \( \cos \theta \) ranges from -1 to 1, as it represents how far a point is horizontally from the center of the circle (0,0).
• When \( \theta \) is 0 or any multiple of \(2\pi\), \( \cos \theta \) equals 1, meaning the point is at the farthest right on the circle.
• Conversely, when \( \theta \) is equal to \(\pi\) or odd multiples of \(\pi\), \( \cos \theta \) is -1, indicating the farthest left.

The key to understanding the cosine function is recognizing that when \( \cos \theta = 0\), it implies that the point on the unit circle lies directly on the y-axis, at angles such as \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). These angles lead to undefined tangent values.

Tangent Function

The tangent function, represented as \( \tan \theta \), is the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship is pivotal in determining scenarios where the tangent is undefined.

Key points to remember about tangent:

• \( \tan \theta \) offers insight into the slope of the line connecting the origin to the point on the unit circle.
• When \( \cos \theta = 0 \), the denominator of this ratio becomes zero, leading to an undefined value of \( \tan \theta \).
• Such undefined cases occur at angles where the point lies on the y-axis of the unit circle, such as \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). At these angles, the steepness or slope of the tangent extends infinitely large, thus undefined.

The lack of definition for tangent at these points is because one cannot divide by zero. Therefore, knowing where cosine is zero is essential for pinpointing undefined tangents.

Unit Circle

The unit circle is a fundamental tool in trigonometry, offering a visual representation of trigonometric functions, including cosine and tangent. Imagine a circle with a radius of 1 in the Cartesian coordinate system.

This visual helps simplify complex trigonometric concepts:

• Any angle \( \theta \) on this circle can be represented as a point \((\cos \theta, \sin \theta)\) on the circumference.
• Key angles on the unit circle include 0, \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \). At these points, coordinates switch between critical values of 0 and 1 or -1.
• The horizontal projection represents \( \cos \theta \), whereas the vertical projection shows \( \sin \theta \).
• When \( \cos \theta = 0 \), the point is aligned vertically with the y-axis, leading to undefined tangent values at these angles.

By understanding the unit circle, students can effortlessly deduce the behavior of sine, cosine, and tangent across different angles, making it a critical learning tool in trigonometry.