Finding Undefined Tangent Values On The Unit Circle

Hey math enthusiasts! Let's dive into a classic trigonometry question. We're going to explore the unit circle and pinpoint when the tangent function goes undefined. It's a fundamental concept, and understanding it unlocks a deeper understanding of trigonometric functions. So, let's get started, shall we?

Unpacking the Tangent Function and the Unit Circle

Alright, before we get to the core of the question, let's quickly recap what the tangent function and the unit circle are all about. This will provide a solid foundation for grasping the concept.

The unit circle is a circle with a radius of 1, centered at the origin (0, 0) of a coordinate plane. It's a handy tool for visualizing and understanding trigonometric functions because it links angles to coordinates. Now, if we take any point on the unit circle, that point can be defined by the coordinates (cos θ, sin θ), where θ represents the angle formed between the positive x-axis and the line connecting the origin to that point.

Now, for the tangent function (tan θ). It is one of the primary trigonometric functions, and it's defined as the ratio of sine to cosine: tan θ = sin θ / cos θ. In simpler terms, if you're on the unit circle, the tangent of an angle θ is essentially the y-coordinate (sin θ) divided by the x-coordinate (cos θ) of the point on the circle that corresponds to that angle. Keep in mind that the tangent function helps us find the slope of the line that passes through the origin and intersects with the unit circle at the given angle. This is super important to remember, as it gives you a visual cue for understanding when the tangent function will be undefined.

Now, the problem comes up when we try to divide by zero. And since we know that tan θ = sin θ / cos θ, then we need to know when cos θ = 0. We'll explore this further in the next section.

Why Tangent is Undefined

As mentioned earlier, the tangent function is undefined when cos θ = 0. This is because we can't divide any number by zero in mathematics. Doing so breaks the rules, creating something that is not mathematically valid.

So, on the unit circle, we need to find the angles where the x-coordinate (which is cos θ) is equal to zero. These are the points on the circle where the line forms a right angle with the x-axis. Thinking about the unit circle, where does the x-coordinate become zero? That happens at the top and bottom of the circle.

• When θ = π/2 (90 degrees), the point on the unit circle is (0, 1). Here, cos θ = 0.
• When θ = 3π/2 (270 degrees), the point on the unit circle is (0, -1). Again, cos θ = 0.

At these specific angles, since cos θ = 0, tan θ is undefined.

Let's break this down even further. Think about the graph of the tangent function. It has vertical asymptotes at these angles, which are lines that the graph gets infinitely close to but never touches. This is another way to visualize that the tangent function is undefined at these points.

So, as a summary, to figure out when tan θ is undefined, we need to look for angles where the x-coordinate (cos θ) is zero. That occurs at π/2 and 3π/2.

Solving the Question Step-by-Step

Alright, let's take a closer look at the question and break it down. We're looking for the angle(s) where tan θ is undefined within the range 0 < θ ≤ 2π. We know that tan θ = sin θ / cos θ. Let's see how each option stacks up.

A. θ = π and θ = 2π

At θ = π, the point on the unit circle is (-1, 0), so cos θ = -1 and sin θ = 0. Therefore, tan θ = 0 / -1 = 0. At θ = 2π, we return to the starting point (1, 0), and cos θ = 1 and sin θ = 0. Therefore, tan θ = 0 / 1 = 0. This option is incorrect because tan θ is not undefined at these angles.

B. sin θ = cos θ

This option looks for angles where the sine and cosine are equal. In other words, where the x and y coordinates on the unit circle are the same. This occurs at θ = π/4 and θ = 5π/4, but tan θ is defined at these points, and sin θ = cos θ. This option is incorrect.

C. θ = π/2 and θ = 3π/2

At θ = π/2, the point on the unit circle is (0, 1), so cos θ = 0. As we know, division by zero is not possible, so tan θ is undefined here. Similarly, at θ = 3π/2, the point is (0, -1), again making cos θ = 0, and tan θ is also undefined. That's the correct answer.

D. sin θ = 1 / cos θ

This expression can be rewritten as tan θ = 1 / cos^2 θ. This is not a direct indication of where tan θ is undefined. The points where sin θ = 1 / cos θ are not where the tangent is undefined. This option is incorrect.

So, the correct answer is C. θ = π/2 and θ = 3π/2. These are the angles where the tangent function is undefined on the unit circle.

Conclusion: Mastering the Tangent's Uncharted Territory

Understanding where trigonometric functions like the tangent are undefined is key to mastering trigonometry. It's like understanding the rules of the road before you start driving. It might seem tricky at first, but with a good grasp of the unit circle and the definition of the tangent function (tan θ = sin θ / cos θ), it becomes straightforward.

Remember, the tangent function is undefined at angles where the cosine is zero. This happens at π/2 and 3π/2 on the unit circle. Keep practicing, keep exploring, and you'll find that trigonometry becomes more intuitive with each step. So, keep exploring the fascinating world of mathematics and trigonometry. Happy calculating, guys!

This is just one question, and there's so much more to discover. Stay curious, keep learning, and don't hesitate to ask questions. You got this!