To solve this, we can refer to the unit circle values for 180°. The angle $180^\circ$ corresponds to the point $(-1, 0)$ on the unit circle.

Now, let’s compute the six trigonometric functions:

1. sin 180°: The sine of an angle is the y-coordinate on the unit circle. $\sin 180^\circ = 0$

2. cos 180°: The cosine of an angle is the x-coordinate on the unit circle. $\cos 180^\circ = -1$

3. tan 180°: The tangent of an angle is the ratio of sine to cosine: $\tan 180^\circ = \frac{\sin 180^\circ}{\cos 180^\circ} = \frac{0}{-1} = 0$

4. cot 180°: The cotangent is the reciprocal of the tangent. $\cot 180^\circ = \frac{1}{\tan 180^\circ} = \frac{1}{0} = \text{UNDEFINED}$

5. sec 180°: The secant is the reciprocal of the cosine. $\sec 180^\circ = \frac{1}{\cos 180^\circ} = \frac{1}{-1} = -1$

6. csc 180°: The cosecant is the reciprocal of the sine. $\csc 180^\circ = \frac{1}{\sin 180^\circ} = \frac{1}{0} = \text{UNDEFINED}$

Here’s the completed table:

• $\sin 180^\circ = 0$
• $\cos 180^\circ = -1$
• $\tan 180^\circ = 0$
• $\cot 180^\circ = \text{UNDEFINED}$
• $\sec 180^\circ = -1$
• $\csc 180^\circ = \text{UNDEFINED}$

Would you like any further clarification or details on these concepts?

Here are five questions related to this topic:

1. What is the significance of the unit circle in trigonometry?
2. How do you determine the coordinates of a point on the unit circle for any given angle?
3. Why are some trigonometric functions undefined for certain angles?
4. How do reciprocal trigonometric functions (like secant, cosecant, and cotangent) relate to the basic trigonometric functions?
5. How would the trigonometric values change for angles in other quadrants?

Tip: Always remember that the tangent and cotangent functions can be undefined at certain angles where the cosine or sine values are zero, because division by zero is undefined.