Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine where [tex]\(\tan \theta\)[/tex] is undefined on the unit circle for [tex]\(0 \leq \theta \leq 2\pi\)[/tex], we need to understand the definition of the tangent function in terms of sine and cosine:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
For the tangent function to be undefined, the denominator (the cosine of the angle) must be zero. Thus, we need to find the values of [tex]\(\theta\)[/tex] for which [tex]\(\cos \theta = 0\)[/tex].
On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point. The cosine function is zero at two specific angles within one full rotation [tex]\([0, 2\pi]\)[/tex]:
1. [tex]\(\theta = \frac{\pi}{2}\)[/tex]
2. [tex]\(\theta = \frac{3\pi}{2}\)[/tex]
At these angles, the sine function [tex]\(\sin \theta\)[/tex] is either [tex]\(1\)[/tex] or [tex]\(-1\)[/tex] respectively, but the cosine function [tex]\(\cos \theta\)[/tex] is zero, making the tangent function [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] undefined.
Therefore, the values of [tex]\(\theta\)[/tex] at which [tex]\(\tan \theta\)[/tex] is undefined are:
[tex]\[ \theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2} \][/tex]
So, the correct answer is [tex]\(\theta=\frac{\pi}{2}\)[/tex] and [tex]\(\theta=\frac{3\pi}{2}\)[/tex].
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
For the tangent function to be undefined, the denominator (the cosine of the angle) must be zero. Thus, we need to find the values of [tex]\(\theta\)[/tex] for which [tex]\(\cos \theta = 0\)[/tex].
On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point. The cosine function is zero at two specific angles within one full rotation [tex]\([0, 2\pi]\)[/tex]:
1. [tex]\(\theta = \frac{\pi}{2}\)[/tex]
2. [tex]\(\theta = \frac{3\pi}{2}\)[/tex]
At these angles, the sine function [tex]\(\sin \theta\)[/tex] is either [tex]\(1\)[/tex] or [tex]\(-1\)[/tex] respectively, but the cosine function [tex]\(\cos \theta\)[/tex] is zero, making the tangent function [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] undefined.
Therefore, the values of [tex]\(\theta\)[/tex] at which [tex]\(\tan \theta\)[/tex] is undefined are:
[tex]\[ \theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{3\pi}{2} \][/tex]
So, the correct answer is [tex]\(\theta=\frac{\pi}{2}\)[/tex] and [tex]\(\theta=\frac{3\pi}{2}\)[/tex].
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.