If cotAcotBcotC>0 the triangle is ?
The cotangent is 1/tan. From your mnemonic (All Students Take Calculus) (that's the one I like, anyway)
you know that in quadrant I, All are positive, in quadrant II the Sine is positive, in quadrant III the Tangent is positive, and in quadrant IV the Cosine is positive.
The only way for the product of the 3 cotangents to be positive and the angles be part of a triangle is for the cotangents to all be positive. This means that the 3 angles need to be in the first quadrant, that is, less than
90 degrees. A triangle with all three angles less than 90 is called an "acute" triangle