Undefined Trigonometric Function At Θ = Π/2 Radians

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of trigonometry to tackle a question that often pops up in exams and problem-solving scenarios: Which trigonometric function is undefined when θ = π/2 radians? If you've ever scratched your head over this, you're in the right place. We're going to break it down step by step, making sure you not only understand the answer but also the why behind it. So, let's get started!

Understanding Radians and the Unit Circle

Before we jump into the specific trigonometric functions, let's quickly recap radians and the unit circle. This foundational knowledge is crucial for understanding why certain functions are undefined at particular angles.

Radian Measure: Radians are a way of measuring angles, just like degrees. However, instead of dividing a circle into 360 degrees, we measure it in terms of π (pi). A full circle is 2π radians, half a circle (180 degrees) is π radians, and a quarter circle (90 degrees) is π/2 radians. So, when we talk about θ = π/2 radians, we're referring to a 90-degree angle.

The Unit Circle: The unit circle is a circle with a radius of 1, centered at the origin (0,0) on the coordinate plane. It's an incredibly useful tool in trigonometry because it allows us to visualize trigonometric functions as coordinates. Any point on the unit circle can be represented as (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line connecting the origin to that point. This representation is fundamental to understanding the behavior of trigonometric functions.

To truly grasp the concept, let's visualize θ = π/2 on the unit circle. Imagine starting at the positive x-axis (0 radians) and rotating counterclockwise until you reach the positive y-axis. That's π/2 radians. The coordinates of this point on the unit circle are (0, 1). Remember, the x-coordinate represents cos θ, and the y-coordinate represents sin θ. Therefore, at θ = π/2, cos(π/2) = 0 and sin(π/2) = 1. Keep this in mind as we explore the trigonometric functions.

Delving into Trigonometric Functions and Their Definitions

Now that we've refreshed our understanding of radians and the unit circle, let's take a closer look at the trigonometric functions themselves. There are six primary trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each of these functions relates the angles of a right triangle to the ratios of its sides. Understanding their definitions is key to determining when they are undefined.

Sine (sin θ): The sine of an angle θ is defined as the ratio of the opposite side to the hypotenuse in a right triangle. On the unit circle, sin θ corresponds to the y-coordinate of the point on the circle. As we established earlier, sin(π/2) = 1.

Cosine (cos θ): The cosine of an angle θ is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. On the unit circle, cos θ corresponds to the x-coordinate of the point on the circle. We know that cos(π/2) = 0.

Tangent (tan θ): The tangent of an angle θ is defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as the ratio of sine to cosine: tan θ = sin θ / cos θ. This is a crucial relationship, as it reveals that the tangent function will be undefined whenever cos θ = 0.

Cosecant (csc θ): The cosecant of an angle θ is the reciprocal of the sine function: csc θ = 1 / sin θ. Therefore, csc θ will be undefined whenever sin θ = 0.

Secant (sec θ): The secant of an angle θ is the reciprocal of the cosine function: sec θ = 1 / cos θ. So, sec θ will be undefined whenever cos θ = 0.

Cotangent (cot θ): The cotangent of an angle θ is the reciprocal of the tangent function: cot θ = 1 / tan θ. It can also be expressed as the ratio of cosine to sine: cot θ = cos θ / sin θ. This means cot θ will be undefined whenever sin θ = 0.

Understanding these definitions and relationships is essential for identifying which function is undefined at θ = π/2. Now, let's apply this knowledge to the specific question at hand.

Identifying the Undefined Function at θ = π/2

Now that we have a solid understanding of trigonometric functions and their definitions, let's pinpoint the function that's undefined when θ = π/2. We'll go through each option systematically:

cos θ: We already know that cos(π/2) = 0. So, the cosine function is defined at θ = π/2.

cot θ: As we discussed, cot θ = cos θ / sin θ. At θ = π/2, cot(π/2) = cos(π/2) / sin(π/2) = 0 / 1 = 0. Therefore, the cotangent function is also defined at θ = π/2.

csc θ: The cosecant function is the reciprocal of the sine function: csc θ = 1 / sin θ. At θ = π/2, sin(π/2) = 1, so csc(π/2) = 1 / 1 = 1. This means the cosecant function is defined at θ = π/2.

tan θ: The tangent function is defined as tan θ = sin θ / cos θ. At θ = π/2, sin(π/2) = 1 and cos(π/2) = 0. Therefore, tan(π/2) = 1 / 0. Division by zero is undefined in mathematics. This is the key to our answer!

Therefore, the tangent function (tan θ) is undefined when θ = π/2 radians.

Why is Division by Zero Undefined?

It's crucial to understand why division by zero is undefined. Think of division as the inverse operation of multiplication. When we say 10 / 2 = 5, we're essentially asking,