Determine all the values of theta,0 degrees < or = theta < 360 satisfying csc(x)<0 and cot(x)>0
csc x < 0 in QIII,QIV
cot x > 0 in QI, QIII
So, all 180 < x < 270 satisfy the requirements.
To determine the values of θ satisfying the given conditions, we need to recall the definitions of csc(x) and cot(x) and the signs of these trigonometric ratios in each quadrant.
The cosecant function, csc(x), is defined as the reciprocal of the sine function: csc(x) = 1/sin(x). A function is negative when its reciprocal is negative. Thus, csc(x) < 0 when sin(x) < 0.
The cotangent function, cot(x), is defined as the reciprocal of the tangent function: cot(x) = 1/tan(x). A function is positive when its reciprocal is positive. Thus, cot(x) > 0 when tan(x) > 0.
Now, let's consider each quadrant and analyze the signs of sin(x) and tan(x):
1. In the first quadrant (0° < θ < 90°), both sin(x) and tan(x) are positive. Thus, this quadrant does not satisfy the conditions csc(x) < 0 and cot(x) > 0.
2. In the second quadrant (90° < θ < 180°), sin(x) is positive and tan(x) is negative. Here, csc(x) < 0 since sin(x) < 0, but cot(x) < 0 because tan(x) < 0. Therefore, this quadrant does not satisfy the conditions.
3. In the third quadrant (180° < θ < 270°), sin(x) is negative and tan(x) is positive. Both conditions are satisfied in this quadrant. So, we get a valid solution range of 180° < θ < 270°.
4. In the fourth quadrant (270° < θ < 360°), sin(x) and tan(x) are both negative. Both conditions are not satisfied in this quadrant.
Therefore, the range of θ satisfying csc(x) < 0 and cot(x) > 0 is 180° ≤ θ < 270°.