If cos<0 and csc<0 then if cos Θ<0 and csc Θ<0, then 270<Θ<360 0<Θ<90 180<Θ<270 90<Θ<180

Question

If cos<0 and csc<0 then if cos Θ<0 and csc Θ<0, then

270<Θ<360

0<Θ<90

180<Θ<270

90<Θ<180

Answer 1

cosine and cosecants of angles are both negative only in quadrant III. both positive only in quadrant I. one negative one positive in quadrants II and IV

the angles are between 180 and 270 degrees

quadrant III has angles only between 180 and 270

or between pi and 3pi/2 radians

Answer 2

Think of the regular coordinate plane. Thinking about left and right on the coordinate plane, x-values that are right of the y-axis are positive while x-values that are left of the y-axis are negative. Thinking about up and down, y-values that are above of the x-axis are positive while y-values that are below of the x-axis are negative. Like this:

The quadrants are marked. Notice that in Quadrant I (QI) that both x-values and y-values are positive, in QII, x-values are negative and y-values are positive. In QIII, both x-values and y-values are negative, and in QIV, x-values are positive and y-values are negative.

In trig, cosine is like the x-value and sine is like the y-value. Since sec = 1/cos and csc = 1/sin, then secant (sec) has the same sign as the x-value and cosecant (csc) has the same sign as the y-value,

So when cos(θ) is negative (< 0) and when csc(θ) is negative (< 0), you must be in QIII which is between 180° and 270°