Question

Determine the quadrant when the terminal side of the angle lies according to the following conditions: tan (t) > 0, csc (t) < 0
Quadrant I

Quadrant III

Quadrant IV

Quadrant II

Answer 1

Answer:  The answer is (C) Quadrant III.

Step-by-step explanation:  We are given to determine the quadrant when the terminal side of the angle lies according to the following conditions: tan (t) > 0, csc (t) < 0.

We know that

In Quadrant I, all the trigonometric ratios are positive (greater than 0).

In Quadrant II, only sine and co-secant ratios are positive.

In Quadrant III, only tangent and cotangent ratios are positive.

In Quadrant IV, only the cosine and secant ratios are positive.

According to the given condition,

tan (t) is positive - Quadrant I or Quadrant III

and

csc (t) is negative - Quadrant III or Quadrant IV.  

Taking the common quadrant, the resulting quadrant will be Quadrant III.

Thus, (B) Quadrant III is the correct option.

Answer 2

Use the CAST rule for positive values in the 4th, 1st, 2nd and 3rd quadrants.
tan(t)>0 => either Q3 or Q1
csc(t)<0 => sin(t)<0 => Q4 or Q3
Since Q3 satisfies both conditions, the given angle lies in Q3