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).