Write cot(t) in terms of sin(t) if the terminal point determined by t is in the first quadrant. Do not leave a fraction inside a radical or use absolute value.

cot(t) = cos(t)/sin(t)

= √(1 - sin^2(t)) / sin(t)

To express cot(t) in terms of sin(t), we can use the trigonometric identity:

cot(t) = 1 / tan(t) = 1 / (sin(t) / cos(t))

In the first quadrant, both sin(t) and cos(t) are positive, so we don't need to use absolute value. By inverting the fraction:

cot(t) = cos(t) / sin(t)

Therefore, cot(t) in terms of sin(t) when the terminal point determined by t is in the first quadrant is cos(t) / sin(t).

To express cot(t) in terms of sin(t), we need to use the trigonometric identity relating cotangent and sine:

cot(t) = 1 / tan(t),

where tan(t) = sin(t) / cos(t).

Since the terminal point determined by t is in the first quadrant, both sine and cosine values are positive. We can substitute these values into the equation to obtain:

cot(t) = 1 / (sin(t) / cos(t)).

Next, we can simplify this expression using the reciprocal of a fraction rule:

cot(t) = cos(t) / sin(t).

Therefore, when the terminal point determined by t is in the first quadrant, cot(t) can be expressed as cos(t) / sin(t).