Final answer:
Given cot(t) = 5, we consider a right triangle with an adjacent side of 5 and an opposite side of 1. We find the hypotenuse using the Pythagorean theorem, and then calculate the other trigonometric functions: sin(t), cos(t), tan(t), sec(t), and csc(t).
Step-by-step explanation:
If given that cot(t) = 5, we can find the other five trigonometric functions by using trigonometric identities and the definition of cotangent. The cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle (cot(t) = adjacent/opposite). Since cot(t) = 5, we can consider the adjacent side to be 5 and the opposite side to be 1, which means we can use the Pythagorean theorem to find the hypotenuse.
Using the Pythagorean theorem, hypotenuse = √(opposite^2 + adjacent^2) = √(1^2 + 5^2) = √(26). Thus, sin(t) = opposite/hypotenuse = 1/√(26), which we can rationalize to √(26)/26. Similarly, cos(t) = adjacent/hypotenuse = 5/√(26), which can be rationalized to 5√(26)/26. From these, we can find tan(t) = sin(t)/cos(t) = 1/5, which is just the reciprocal of cot(t). The secant is the reciprocal of the cosine, so sec(t) = 1/cos(t) = √(26)/5. And the cosecant is the reciprocal of the sine, so csc(t) = 1/sin(t) = √(26).