Question

identify the quadrant or quadrants for the angles satisfying the following conditions. csc α < 0 and sec α > 0 the angles α satisfying the given conditions are in quadrant(s) . (type 1, 2, 3, or 4. use a comma to separate answers as needed.)

Explanation:

Step1: Recall reciprocal identities

Recall that \(\csc\alpha=\frac{1}{\sin\alpha}\) and \(\sec\alpha = \frac{1}{\cos\alpha}\). So, \(\csc\alpha<0\) implies \(\frac{1}{\sin\alpha}<0\), which means \(\sin\alpha<0\) (since the reciprocal of a number has the same sign as the number itself, except when the number is zero, but \(\sin\alpha\) can't be zero here as \(\csc\alpha\) would be undefined). Similarly, \(\sec\alpha>0\) implies \(\frac{1}{\cos\alpha}>0\), so \(\cos\alpha>0\) (again, \(\cos\alpha\) can't be zero here as \(\sec\alpha\) would be undefined).

Step2: Determine quadrants for sine and cosine

• Sine is negative (\(\sin\alpha<0\)) in Quadrants III and IV (because in Quadrant I, \(\sin\) is positive; Quadrant II, \(\sin\) is positive; Quadrant III, \(\sin\) is negative; Quadrant IV, \(\sin\) is negative).
• Cosine is positive (\(\cos\alpha>0\)) in Quadrants I and IV (because in Quadrant I, \(\cos\) is positive; Quadrant II, \(\cos\) is negative; Quadrant III, \(\cos\) is negative; Quadrant IV, \(\cos\) is positive).

Step3: Find the intersection of quadrants

We need the quadrant where \(\sin\alpha<0\) and \(\cos\alpha>0\) simultaneously. The intersection of Quadrants III, IV (where \(\sin\alpha<0\)) and Quadrants I, IV (where \(\cos\alpha>0\)) is Quadrant IV.

Answer:

4