What Is Theta When Cos Theta Equals I/i?

When you encounter a mathematical problem that seems a bit unusual, like the expression $\cos \theta = \frac{i}{i}$, it's important to break it down and understand what each part means. Let's dive into this specific scenario and figure out what $\theta$ could be, exploring the different options provided. We'll look at trigonometric functions, complex numbers, and inverse trigonometric functions to shed light on this intriguing question. By the end, you'll have a clearer understanding of how to approach such problems and what the correct representation of $\theta$ is.

Understanding the Expression $\cos \theta = \frac{i}{i}$

Let's start by dissecting the given equation: $\cos \theta = \frac{i}{i}$. The first thing that catches the eye is the expression $\frac{i}{i}$. In mathematics, 'i' typically represents the imaginary unit, which is defined as the square root of -1 ($i^2 = -1$). However, in this specific context, the expression $\frac{i}{i}$ simplifies to 1, as any non-zero number divided by itself equals 1. Therefore, the equation effectively becomes $\cos \theta = 1$. This is a fundamental trigonometric identity. The cosine function, $\cos \theta$, represents the x-coordinate of a point on the unit circle corresponding to an angle $\theta$ measured from the positive x-axis. For $\cos \theta$ to equal 1, the point on the unit circle must be at (1, 0). This occurs when the angle $\theta$ is 0 radians, or any integer multiple of $2\pi$ radians (i.e., $0, \pm 2\pi, \pm 4\pi$, and so on). So, we are looking for an expression that represents an angle whose cosine is 1.

Analyzing the Options for $\theta$

Now, let's examine the given options to see which one correctly represents an angle $\theta$ such that $\cos \theta = 1$. The options provided are:

A. $\sin \left(\frac{\pi}{6}\right)$ B. $\cos \left(\frac{\pi}{6}\right)$ C. $\arccos \left(\frac{\pi}{6}\right)$ D. $\arcsin \left(\frac{\pi}{6}\right)$

We need to evaluate each of these expressions to determine if they result in an angle whose cosine is 1.

Option A: $\sin \left(\frac{\pi}{6}\right)$

The expression $\sin \left(\frac{\pi}{6}\right)$ involves the sine of $\frac{\pi}{6}$ radians (which is 30 degrees). The value of $\sin \left(\frac{\pi}{6}\right)$ is $\frac{1}{2}$. So, if $\theta = \frac{1}{2}$, then $\cos \theta = \cos \left(\frac{1}{2}\right)$. This value is not 1. Therefore, option A is incorrect.

Option B: $\cos \left(\frac{\pi}{6}\right)$

This expression, $\cos \left(\frac{\pi}{6}\right)$, gives the cosine of $\frac{\pi}{6}$ radians. The value of $\cos \left(\frac{\pi}{6}\right)$ is $\frac{\sqrt{3}}{2}$. This is not equal to 1. So, if $\theta = \frac{\sqrt{3}}{2}$, then $\cos \theta = \cos \left(\frac{\sqrt{3}}{2}\right)$, which is not 1. Therefore, option B is incorrect.

Option C: $\arccos \left(\frac{\pi}{6}\right)$

The expression $\arccos \left(\frac{\pi}{6}\right)$ represents the angle whose cosine is $\frac{\pi}{6}$. The value of $\frac{\pi}{6}$ is approximately $\frac{3.14159}{6} \approx 0.5236$. The arccosine function, $\arccos(x)$, returns an angle in the range $[0, \pi]$. For $\arccos(x)$ to represent an angle $\theta$ such that $\cos \theta = 1$, the input to the arccosine function must be 1. Since $\frac{\pi}{6} \neq 1$, this option does not directly provide an angle whose cosine is 1. In fact, $\arccos \left(\frac{\pi}{6}\right)$ will give an angle whose cosine is approximately 0.5236, not 1.

Option D: $\arcsin \left(\frac{\pi}{6}\right)$

Similarly, $\arcsin \left(\frac{\pi}{6}\right)$ represents the angle whose sine is $\frac{\pi}{6}$. The arcsine function, $\arcsin(x)$, returns an angle in the range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. If $\theta = \arcsin \left(\frac{\pi}{6}\right)$, then $\sin \theta = \frac{\pi}{6}$. We are looking for an angle $\theta$ such that $\cos \theta = 1$. Using the identity $\sin^2 \theta + \cos^2 \theta = 1$, if $\cos \theta = 1$, then $\sin^2 \theta = 1 - 1^2 = 0$, which means $\sin \theta = 0$. Since $\frac{\pi}{6} \neq 0$, this option also does not lead to an angle whose cosine is 1.

Re-evaluating the Problem and Options

It seems there might be a misunderstanding or a typo in how the question is framed or the options are presented, as none of the given options directly evaluate to an angle whose cosine is 1. Let's revisit the core of the problem: $\cos \theta = 1$. The principal value for $\theta$ that satisfies this equation is $\theta = 0$. If we are looking for any angle, then $\theta = 2n\pi$ for any integer $n$.

However, if we are forced to choose from the given options, we must consider what each option represents in relation to the value 1. The equation $\cos \theta = \frac{i}{i}$ simplifies to $\cos \theta = 1$. We need to find an expression that, when evaluated, results in an angle $\theta$ such that $\cos \theta = 1$. The primary angle for which $\cos \theta = 1$ is $\theta = 0$. None of the options directly evaluate to 0.

Let's consider the possibility that the question is asking which expression represents $\theta$, and that $\theta$ itself is related to one of these values. If $\cos \theta = 1$, then $\theta = 0, 2\pi, 4\pi, ...$. None of the options are directly equal to these values.

There might be a confusion between the value of $\cos \theta$ and the angle $\theta$. The problem states $\cos \theta = \frac{i}{i} = 1$. We are looking for an expression that represents $\theta$.

Let's re-examine the options as potential values for $\theta$ itself, and then check if their cosine is 1. This doesn't seem to be the intent, as the options are functions of $\frac{\pi}{6}$.

There's a critical interpretation here: The question asks