1) Determine the quadrant in which Q lies.
a) sec Q > 0 and cot Q < 0
I know that sec is in quadrant 4 and cot is in 3. Now how do I find the quadrant?
b) csc Q < 0 and tan Q > 0
csc is in quadrant 2 and tan is 3.
Do them this way:
a) sec Q > 0 and cot Q < 0
Since the signs of secant and cotangents are the same as the signs of cosine and tangent we could say
cos Q > 0 and tan Q < 0
or
the cos Q is positive and tan Q is negative.
so Q could be in either I or IV and
Q could be in either II or IV
so it looks like Q has to be in IV.
Do the other one the same way.
To determine the quadrant in which angle Q lies, you need to consider the signs of the trigonometric functions involved.
a) In this case, you are given that sec(Q) > 0 and cot(Q) < 0.
Secant is positive in Quadrants I and IV, and cotangent is negative in Quadrants II and IV.
Since sec(Q) is positive (greater than 0), it means that Q lies in Quadrant I or IV.
And since cot(Q) is negative (less than 0), it means that Q lies in Quadrant II or IV.
Therefore, the possible quadrants for Q are Quadrant IV (because secant is positive) or Quadrant II (because cotangent is negative).
To determine the exact quadrant, you need additional information about the relationship between sec(Q) and cot(Q). If you have that information, you can use it to narrow down the possibilities and determine the specific quadrant in which Q lies.
b) In this case, you are given that csc(Q) < 0 and tan(Q) > 0.
Cosecant is negative in Quadrants III and IV, and tangent is positive in Quadrants I and III.
Since csc(Q) is negative (less than 0), it means that Q lies in Quadrant III or IV.
And since tan(Q) is positive (greater than 0), it means that Q lies in Quadrant I or III.
Therefore, the possible quadrants for Q are Quadrant I (because tangent is positive) or Quadrant III (because cosecant is negative).
Again, to determine the exact quadrant, you need additional information about the relationship between csc(Q) and tan(Q). If you have that information, you can use it to narrow down the possibilities and determine the specific quadrant in which Q lies.