Find the values of the trigonometric functions of t from the given information.csc(t) = 6,    cos(t) < 0sin(t) = cos(t) = tan(t) = sec(t) = cot(t) =

Given that csc(t) = 6, we can find sin(t) as the reciprocal of csc(t). So, sin(t) = 1/csc(t) = 1/6.

Since cos(t) < 0, we know that t is in the second or third quadrant (where cosine is negative).

In the second quadrant, sine is positive, so we can assume t is in the second quadrant because sin(t) is positive.

Now, we can find the other trigonometric functions.

cos(t) can be found using the Pythagorean identity sin²(t) + cos²(t) = 1. Solving for cos(t), we get cos(t) = sqrt(1 - sin²(t)) = sqrt(1 - (1/6)²) = sqrt(1 - 1/36) = sqrt(35/36). Since we're in the second quadrant, cos(t) is negative, so cos(t) = -sqrt(35)/6.

tan(t) = sin(t)/cos(t) = (1/6) / (-sqrt(35)/6) = -1/sqrt(35).

sec(t) is the reciprocal of cos(t), so sec(t) = -6/sqrt(35).

cot(t) is the reciprocal of tan(t), so cot(t) = -sqrt(35).