To evaluate the angle $\frac{3\pi}{2}$ using the six circular functions, we can start by identifying the position of this angle on the unit circle.

1. Angle in Degrees: $\frac{3\pi}{2} \text{ radians} = 270^\circ$ This angle points directly downward on the unit circle.

2. Coordinates on the Unit Circle: The coordinates of the point on the unit circle corresponding to $\frac{3\pi}{2}$ are: $(0, -1)$

3. Circular Functions: The six circular functions for this angle are:

   • Sine: $\sin\left(\frac{3\pi}{2}\right) = -1$

   • Cosine: $\cos\left(\frac{3\pi}{2}\right) = 0$

   • Tangent: $\tan\left(\frac{3\pi}{2}\right) = \frac{\sin\left(\frac{3\pi}{2}\right)}{\cos\left(\frac{3\pi}{2}\right)} = \frac{-1}{0} \text{ (undefined)}$

   • Cosecant (reciprocal of sine): $\csc\left(\frac{3\pi}{2}\right) = \frac{1}{\sin\left(\frac{3\pi}{2}\right)} = \frac{1}{-1} = -1$

   • Secant (reciprocal of cosine): $\sec\left(\frac{3\pi}{2}\right) = \frac{1}{\cos\left(\frac{3\pi}{2}\right)} = \frac{1}{0} \text{ (undefined)}$

   • Cotangent (reciprocal of tangent): $\cot\left(\frac{3\pi}{2}\right) = \frac{\cos\left(\frac{3\pi}{2}\right)}{\sin\left(\frac{3\pi}{2}\right)} = \frac{0}{-1} = 0$

4. Summary of the Circular Functions:

   • $\sin\left(\frac{3\pi}{2}\right) = -1$
   • $\cos\left(\frac{3\pi}{2}\right) = 0$
   • $\tan\left(\frac{3\pi}{2}\right) = \text{undefined}$
   • $\csc\left(\frac{3\pi}{2}\right) = -1$
   • $\sec\left(\frac{3\pi}{2}\right) = \text{undefined}$
   • $\cot\left(\frac{3\pi}{2}\right) = 0$

If you have any questions or need further details, feel free to ask!

Related Questions:

1. What are the values of the circular functions for $\pi$ radians?
2. How do you convert radians to degrees?
3. Can you explain the unit circle and its significance in trigonometry?
4. What is the periodicity of the sine and cosine functions?
5. How do you determine the signs of the circular functions in different quadrants?

Tip: Always remember the unit circle helps in quickly identifying the values of the circular functions based on the angle's position.