Trig Functions: Which Ones Are Undefined?

Hey math whizzes! Today, we're diving deep into the wild world of trigonometric functions, specifically focusing on when things get a little... undefined. You know, those moments when you try to calculate a value and BAM! You get an error. We've got a few options here, and we need to pick out the two that result in an undefined value. Let's break it down, shall we?

Understanding Undefined in Trigonometry

Alright guys, so what makes a trigonometric function undefined? In simple terms, it's usually when you're trying to divide by zero. Think about the definitions of tangent, cotangent, secant, and cosecant. They're all based on sine and cosine, and sometimes those denominators end up being zero. For instance, remember that $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and $\cot(x) = \frac{\cos(x)}{\sin(x)}$. See that division? If the denominator is zero, we've got a problem – an undefined situation. Similarly, $\sec(x) = \frac{1}{\cos(x)}$ and $\csc(x) = \frac{1}{\sin(x)}$. So, if $\cos(x) = 0$, secant is undefined. If $\sin(x) = 0$, cosecant is undefined. It all comes down to the unit circle and where our angle lands us. We're looking for angles where the x-coordinate (for cosine) or the y-coordinate (for sine) is zero. This happens at multiples of $\pi$ for sine (0, $\pi$, $2\pi$, etc.) and multiples of $\frac{\pi}{2}$ for cosine ($\frac{\pi}{2}$, $\frac{3\pi}{2}$, $\frac{5\pi}{2}$, etc.). So, our mission, should we choose to accept it, is to identify which of the given expressions will lead us to divide by zero.

Analyzing the Options

Let's tackle each option one by one, shall we? This is where we put our trigonometry knowledge to the test and figure out which expressions are mathematically impossible to solve. We'll be looking at the unit circle and recalling the values of sine and cosine for these specific angles. Remember, $\sin(\theta)$ is the y-coordinate and $\cos(\theta)$ is the x-coordinate of the point where the terminal side of the angle $\theta$ intersects the unit circle. This visual helps a ton when we're dealing with angles that are multiples of $\frac{\pi}{2}$ or $\pi$. It's like having a cheat sheet, but way cooler because it's pure math! Keep a close eye on the signs as well, because even though we're primarily worried about zero, the quadrant matters for other calculations. But for undefined, it's strictly about that zero denominator. Let's get down to business and check each one.

Option 1: $\csc \left(-\frac{3 \pi}{2}\right)$

First up, we have $\csc \left(-\frac{3 \pi}{2}\right)$. Cosecant is $\frac{1}{\sin(x)}$. So, we need to find the value of $\sin \left(-\frac{3 \pi}{2}\right)$. Now, $-\frac{3 \pi}{2}$ is the same angle as $\frac{\pi}{2}$ (just going in the opposite direction). If you picture the unit circle, starting from the positive x-axis and moving counter-clockwise, $\frac{\pi}{2}$ lands you straight up on the positive y-axis. The coordinates at this point are $(0, 1)$. Therefore, $\sin \left(-\frac{3 \pi}{2}\right) = \sin \left(\frac{\pi}{2}\right) = 1$. Since the sine value is 1 (not zero), $\csc \left(-\frac{3 \pi}{2}\right) = \frac{1}{1} = 1$. This one is defined.

Option 2: $\cot \left(\frac{7 \pi}{2}\right)$

Next, let's check out $\cot \left(\frac{7 \pi}{2}\right)$. Cotangent is $\frac{\cos(x)}{\sin(x)}$. We need to evaluate $\sin \left(\frac{7 \pi}{2}\right)$ and $\cos \left(\frac{7 \pi}{2}\right)$. The angle $\frac{7 \pi}{2}$ is equivalent to $\frac{7 \pi}{2} = \frac{4 \pi}{2} + \frac{3 \pi}{2} = 2\pi + \frac{3 \pi}{2}$. So, it's the same as $\frac{3 \pi}{2}$. On the unit circle, $\frac{3 \pi}{2}$ points straight down along the negative y-axis. The coordinates here are $(0, -1)$. Therefore, $\cos \left(\frac{7 \pi}{2}\right) = 0$ and $\sin \left(\frac{7 \pi}{2}\right) = -1$. Since the sine value is -1 (not zero), $\cot \left(\frac{7 \pi}{2}\right) = \frac{0}{-1} = 0$. This one is also defined.

Option 3: $\csc (3 \pi)$

Moving on to $\csc (3 \pi)$. Again, cosecant is $\frac{1}{\sin(x)}$. We need $\sin(3 \pi)$. The angle $3\pi$ is equivalent to $2\pi + \pi$, so it's the same as $\pi$. On the unit circle, $\pi$ lies on the negative x-axis. The coordinates are $(-1, 0)$. Therefore, $\sin(3 \pi) = 0$. Uh oh! We have a sine value of zero. This means $\csc (3 \pi) = \frac{1}{0}$. Division by zero! This expression is undefined. So, this is one of our answers, guys!

Option 4: $\cot \left(\frac{5 \pi}{3}\right)$

Let's analyze $\cot \left(\frac{5 \pi}{3}\right)$. This is $\frac{\cos(x)}{\sin(x)}$. The angle $\frac{5 \pi}{3}$ is in the fourth quadrant. To find its sine and cosine, we can use the reference angle, which is $\frac{\pi}{3}$. In the fourth quadrant, cosine is positive and sine is negative. So, $\cos \left(\frac{5 \pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$ and $\sin \left(\frac{5 \pi}{3}\right) = -\sin \left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$. Since neither the sine nor the cosine is zero, the cotangent is $\cot \left(\frac{5 \pi}{3}\right) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}$. This one is defined.

Option 5: $\sec \left(-\frac{\pi}{2}\right)$

Finally, we have $\sec \left(-\frac{\pi}{2}\right)$. Secant is $\frac{1}{\cos(x)}$. We need to find $\cos \left(-\frac{\pi}{2}\right)$. The angle $-\frac{\pi}{2}$ is the same as $\frac{3 \pi}{2}$ (going clockwise from the positive x-axis). On the unit circle, this lands us on the negative y-axis at $(0, -1)$. Therefore, $\cos \left(-\frac{\pi}{2}\right) = 0$. Boom! Another division by zero situation: $\sec \left(-\frac{\pi}{2}\right) = \frac{1}{0}$. This expression is also undefined!

The Verdict

So, after carefully examining each option, we found two expressions that lead to division by zero, making them undefined. These are:

• $\csc (3 \pi)$ (because $\sin(3\pi) = 0$)
• $\sec \left(-\frac{\pi}{2}\right)$ (because $\cos\left(-\frac{\pi}{2}\right) = 0$)

There you have it, folks! Always remember to check those denominators when dealing with secant and cosecant. Happy calculating!