The image you provided involves finding the exact values of the six trigonometric functions for a given angle, based on a right triangle inscribed in the unit circle. The values are related to the triangle where:

• The radius (hypotenuse) of the unit circle is $13$,
• One leg is $5$,
• The other leg is $12$.

To determine the six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—we use the relationships from a right triangle:

1. Sine (sin) is the ratio of the opposite side to the hypotenuse:
   $\sin(\theta) = \frac{12}{13}$

2. Cosine (cos) is the ratio of the adjacent side to the hypotenuse:
   $\cos(\theta) = \frac{5}{13}$

3. Tangent (tan) is the ratio of the opposite side to the adjacent side:
   $\tan(\theta) = \frac{12}{5}$

4. Cosecant (csc) is the reciprocal of sine:
   $\csc(\theta) = \frac{13}{12}$

5. Secant (sec) is the reciprocal of cosine:
   $\sec(\theta) = \frac{13}{5}$

6. Cotangent (cot) is the reciprocal of tangent:
   $\cot(\theta) = \frac{5}{12}$

These are the exact values of the six trigonometric functions for the angle $\theta$ formed in the given triangle. If you have any further questions or need clarification, feel free to ask!

Would you like more details on any of the functions or explanations about how these relationships are derived?

Follow-up Questions:

1. How are trigonometric identities like Pythagorean identities used in this context?
2. Can we apply these same principles to an angle in a non-unit circle?
3. What happens to the values of these functions if the sides of the triangle change?
4. How would you solve for angles using trigonometric functions?
5. Can you derive the functions for a given triangle with different side lengths?

Tip: Always visualize the triangle for a better understanding of trigonometric relationships!