The problem asks to express the given trigonometric ratios in terms of an acute angle (less than 90°).

To do this, we use reference angles and apply the trigonometric identities for angles in various quadrants. The basic rule is:

• In quadrant II (angles between 90° and 180°), sine is positive, and cosine, secant, and tangent are negative.
• In quadrant III (angles between 180° and 270°), tangent and cotangent are positive, while sine, cosine, secant, and cosecant are negative.

Let’s express each angle's trigonometric function as an acute angle (reference angle):

1. $\sin 132^\circ$

   • Reference angle: $180^\circ - 132^\circ = 48^\circ$
   • $\sin 132^\circ = \sin 48^\circ$

2. $\cos 175^\circ$

   • Reference angle: $180^\circ - 175^\circ = 5^\circ$
   • $\cos 175^\circ = -\cos 5^\circ$

3. $\csc 156^\circ$

   • Reference angle: $180^\circ - 156^\circ = 24^\circ$
   • $\csc 156^\circ = \csc 24^\circ$

4. $\sec 143^\circ$

   • Reference angle: $180^\circ - 143^\circ = 37^\circ$
   • $\sec 143^\circ = -\sec 37^\circ$

5. $\tan 152^\circ$

   • Reference angle: $180^\circ - 152^\circ = 28^\circ$
   • $\tan 152^\circ = -\tan 28^\circ$

6. $\cos 172^\circ$

   • Reference angle: $180^\circ - 172^\circ = 8^\circ$
   • $\cos 172^\circ = -\cos 8^\circ$

7. $\tan 137^\circ$

   • Reference angle: $180^\circ - 137^\circ = 43^\circ$
   • $\tan 137^\circ = -\tan 43^\circ$

8. $\sin 121^\circ$

   • Reference angle: $180^\circ - 121^\circ = 59^\circ$
   • $\sin 121^\circ = \sin 59^\circ$

9. $\sec 122^\circ$

   • Reference angle: $180^\circ - 122^\circ = 58^\circ$
   • $\sec 122^\circ = -\sec 58^\circ$

10. $\tan 139^\circ$

    • Reference angle: $180^\circ - 139^\circ = 41^\circ$
    • $\tan 139^\circ = -\tan 41^\circ$

11. $\sec 164^\circ$

    • Reference angle: $180^\circ - 164^\circ = 16^\circ$
    • $\sec 164^\circ = -\sec 16^\circ$

12. $\csc 161^\circ$

    • Reference angle: $180^\circ - 161^\circ = 19^\circ$
    • $\csc 161^\circ = \csc 19^\circ$

Would you like more details on any of these? Here are some related questions:

1. How do you determine the sign of trigonometric functions in different quadrants?
2. What are reference angles, and how are they used in trigonometry?
3. Can you explain the symmetry of trigonometric functions around the x-axis and y-axis?
4. How can the CAST rule be applied in these situations?
5. What is the difference between co-functions like secant and cosecant?

Tip: When working with trigonometric angles, always check the quadrant first to determine the sign of the function.