The trigonometric functions of $0$ (radians) can be derived directly from the unit circle or known trigonometric identities. Here are the exact values of the six trigonometric functions at $0$ radians:

1. Sine (sin): $\sin(0) = 0$

2. Cosine (cos): $\cos(0) = 1$

3. Tangent (tan): $\tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0$

4. Cosecant (csc): $\csc(0) = \frac{1}{\sin(0)} = \frac{1}{0}$ This is undefined because division by zero is not defined.

5. Secant (sec): $\sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1$

6. Cotangent (cot): $\cot(0) = \frac{\cos(0)}{\sin(0)} = \frac{1}{0}$ This is also undefined because division by zero is not defined.

So, summarizing:

• $\sin(0) = 0$
• $\cos(0) = 1$
• $\tan(0) = 0$
• $\csc(0)$ is undefined.
• $\sec(0) = 1$
• $\cot(0)$ is undefined.

Would you like more details or have any other questions?

Here are 8 questions that might expand your understanding of this topic:

1. Why is the cosecant of 0 undefined?
2. What is the significance of the unit circle in trigonometry?
3. How do you derive the values of trigonometric functions from the unit circle?
4. Can you explain the relationship between the sine and cosine functions?
5. What are the values of the trigonometric functions at $\pi/2$ radians?
6. How do the trigonometric functions change as the angle increases from $0$ to $2\pi$ radians?
7. What are the trigonometric identities involving sine and cosine?
8. How do you use the Pythagorean identity to verify the values of trigonometric functions?

Tip: Memorizing the values of trigonometric functions at key angles (0, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, etc.) can greatly help in solving trigonometric problems quickly and accurately.