Tan Theta Undefined On Unit Circle: Find Theta

When dealing with trigonometric functions on the unit circle, understanding where these functions are undefined is just as crucial as knowing where they are defined. For the tangent function, $\tan \theta$, this occurs at specific angles where the cosine value is zero. Let's dive deep into the unit circle and explore when $\tan \theta$ is undefined, specifically for angles $0 < \theta \leq 2 \pi$. The tangent of an angle $\theta$ is defined as the ratio of the sine of the angle to the cosine of the angle: $\tan \theta = \frac{\sin \theta}{\cos \theta}$. For $\tan \theta$ to be undefined, the denominator, $\cos \theta$, must be equal to zero. On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. Therefore, we are looking for angles where the x-coordinate is zero.

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. Angles are measured counterclockwise from the positive x-axis. Key angles on the unit circle have easily identifiable sine and cosine values. We're focusing on the interval $0 < \theta \leq 2 \pi$, which represents one full rotation around the circle, excluding the starting point $\theta=0$ but including the ending point $\theta=2 \pi$. We need to find the angles within this range where $\cos \theta = 0$. The points on the unit circle where the x-coordinate is zero are at the top and bottom of the circle, corresponding to the y-axis. These points are (0, 1) and (0, -1). The angle that corresponds to the point (0, 1) is $\theta = \frac{\pi}{2}$ (90 degrees). At this angle, $\sin \frac{\pi}{2} = 1$ and $\cos \frac{\pi}{2} = 0$. Thus, $\tan \frac{\pi}{2} = \frac{1}{0}$, which is undefined. The angle that corresponds to the point (0, -1) is $\theta = \frac{3\pi}{2}$ (270 degrees). At this angle, $\sin \frac{3\pi}{2} = -1$ and $\cos \frac{3\pi}{2} = 0$. Thus, $\tan \frac{3\pi}{2} = \frac{-1}{0}$, which is also undefined. Therefore, within the specified range of $0 < \theta \leq 2 \pi$, the tangent function is undefined at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$. These are the only angles in this interval where the cosine is zero.

Let's consider the options provided to solidify our understanding. Option A suggests $\theta=\pi$ and $\theta=2 \pi$. At $\theta = \pi$, $\cos \pi = -1$ and $\sin \pi = 0$. So, $\tan \pi = \frac{0}{-1} = 0$, which is defined. At $\theta = 2 \pi$, $\cos 2 \pi = 1$ and $\sin 2 \pi = 0$. So, $\tan 2 \pi = \frac{0}{1} = 0$, which is also defined. Thus, option A is incorrect. Option B states $\sin \theta = \cos \theta$. This condition is met when $\tan \theta = 1$. This happens at $\theta = \frac{\pi}{4}$ and $\theta = \frac{5\pi}{4}$ in the given interval. At these angles, the tangent function is defined (it's equal to 1). Therefore, option B does not identify where $\tan \theta$ is undefined. Option D, $\sin \theta = \frac{1}{\cos \theta}$, is equivalent to $\sin \theta \cos \theta = 1$. This is a less common condition to analyze directly for undefined tangents. However, if we consider the fundamental definition of tangent, $\tan \theta = \frac{\sin \theta}{\cos \theta}$, we are looking for cases where $\cos \theta = 0$. Option C presents $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$. As we've already established, at these two angles, the cosine value is 0, making the tangent function undefined. This aligns perfectly with our analysis.

The Core Concept: Division by Zero

The fundamental reason why $\tan \theta$ becomes undefined is rooted in the very definition of division. In mathematics, division by zero is an undefined operation. Since $\tan \theta$ is expressed as $\frac{\sin \theta}{\cos \theta}$, any angle $\theta$ that results in $\cos \theta = 0$ will lead to an undefined tangent. On the unit circle, the cosine of an angle is represented by the x-coordinate of the point where the angle's terminal side intersects the circle. Therefore, we are searching for points on the unit circle where the x-coordinate is zero. These points are located directly on the y-axis. The y-axis intersects the unit circle at two points: (0, 1) and (0, -1). The angle that leads to the point (0, 1) is $\frac{\pi}{2}$ radians (or 90 degrees). At this angle, $\sin \frac{\pi}{2} = 1$ and $\cos \frac{\pi}{2} = 0$. Consequently, $\tan \frac{\pi}{2} = \frac{1}{0}$, which is undefined. The angle that leads to the point (0, -1) is $\frac{3\pi}{2}$ radians (or 270 degrees). At this angle, $\sin \frac{3\pi}{2} = -1$ and $\cos \frac{3\pi}{2} = 0$. Consequently, $\tan \frac{3\pi}{2} = \frac{-1}{0}$, which is also undefined. These are the only two instances within a standard $0$ to $2\pi$ rotation where the x-coordinate on the unit circle is zero.

It's important to remember the domain of the tangent function. The domain of $\tan \theta$ consists of all real numbers except for odd multiples of $\frac{\pi}{2}$. This means $\theta \neq \frac{\pi}{2} + n\pi$, where $n$ is any integer. For the interval $0 < \theta \leq 2 \pi$, the values of $\theta$ that fit this exclusion are precisely $\frac{\pi}{2}$ (when $n=0$) and $\frac{3\pi}{2}$ (when $n=1$). Any other value of $\theta$ in this interval will have a non-zero cosine, and thus a defined tangent value. For instance, at $\theta = \pi$, $\cos \pi = -1$, so $\tan \pi = 0$. At $\theta = 2\pi$, $\cos 2\pi = 1$, so $\tan 2\pi = 0$. These angles result in a tangent value of 0, not an undefined value. The question specifically asks for when $\tan \theta$ is undefined, which strictly means when the denominator of its definition is zero.

Visualizing Undefined Tangents on the Unit Circle

To truly grasp why $\tan \theta$ is undefined at $\frac{\pi}{2}$ and $\frac{3\pi}{2}$, let's visualize it. Imagine the unit circle. The angle $\theta=0$ starts at the point (1, 0) on the right side of the x-axis. As $\theta$ increases counterclockwise, the point on the circle moves. When $\theta$ reaches $\frac{\pi}{2}$, the point on the unit circle is at (0, 1), directly above the origin on the positive y-axis. At this exact position, the line segment from the origin to the point has a slope. The slope of this line segment is $\frac{\Delta y}{\Delta x} = \frac{1-0}{0-0}$, which is undefined because the change in x is zero. Remember that $\tan \theta$ represents the slope of the line segment from the origin to the point $(\cos \theta, \sin \theta)$ on the unit circle. Thus, at $\frac{\pi}{2}$, the slope is undefined. Similarly, when $\theta$ reaches $\frac{3\pi}{2}$, the point on the unit circle is at (0, -1), directly below the origin on the negative y-axis. The slope of the line segment from the origin to (0, -1) is $\frac{-1-0}{0-0}$, again undefined due to a zero change in x. Therefore, the angles $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ are where the tangent function is undefined.

Consider the behavior of $\tan \theta$ as $\theta$ approaches these values. As $\theta$ approaches $\frac{\pi}{2}$ from the left (values slightly less than $\frac{\pi}{2}$), $\sin \theta$ is positive and approaches 1, while $\cos \theta$ is positive and approaches 0 from the positive side. Thus, $\tan \theta = \frac{\sin \theta}{\cos \theta}$ approaches $+\infty$. As $\theta$ approaches $\frac{\pi}{2}$ from the right (values slightly greater than $\frac{\pi}{2}$), $\sin \theta$ is positive and approaches 1, while $\cos \theta$ is negative and approaches 0 from the negative side. Thus, $\tan \theta$ approaches $-\infty$. This asymptotic behavior around $\theta = \frac{\pi}{2}$ is characteristic of an undefined function. The same pattern occurs around $\theta = \frac{3\pi}{2}$, where the tangent approaches $+\infty$ from the left and $-\infty$ from the right.

Why Other Options Are Incorrect

Let's reiterate why the other options are incorrect, reinforcing our choice of C. Option A, $\theta=\pi$ and $\theta=2 \pi$, are angles where the tangent is zero. At $\pi$, we are at the point (-1, 0) on the unit circle. $\tan \pi = \frac{\sin \pi}{\cos \pi} = \frac{0}{-1} = 0$. At $2\pi$, we are back at the point (1, 0). $\tan 2\pi = \frac{\sin 2\pi}{\cos 2\pi} = \frac{0}{1} = 0$. Both are clearly defined values.

Option B, $\sin \theta = \cos \theta$, is a condition where $\tan \theta = 1$. This occurs in the first quadrant at $\theta = \frac{\pi}{4}$ and in the third quadrant at $\theta = \frac{5\pi}{4}$. At these angles, the tangent is 1, which is a perfectly defined number. Option D, $\sin \theta = \frac{1}{\cos \theta}$, can be rewritten as $\sin \theta \cos \theta = 1$. This equation does not directly relate to where the tangent is undefined. To find undefined tangents, we must always focus on the condition $\cos \theta = 0$. The values $\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$ are the only angles in the interval $0 < \theta \leq 2 \pi$ where $\cos \theta$ equals zero. Therefore, these are the only angles where $\tan \theta$ is undefined.

In conclusion, when examining the unit circle for angles $0 < \theta \leq 2 \pi$, the tangent function, $\tan \theta$, is undefined precisely when its denominator, $\cos \theta$, equals zero. This occurs at the angles $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$. These angles correspond to the points on the unit circle where the x-coordinate is zero, leading to division by zero in the definition of tangent. Understanding these critical points is fundamental for analyzing the behavior and domain of trigonometric functions.

For further exploration into trigonometric functions and their properties, you can visit Khan Academy's Trigonometry Section for comprehensive lessons and practice problems.