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) \boxed{}.
(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 \(\sin\alpha>0\) (since the reciprocal of a positive number is positive), and \(\sec\alpha>0\) implies \(\cos\alpha>0\) (for the same reason).

Step2: Determine quadrants for sine and cosine

• Sine is positive (\(\sin\alpha>0\)) in Quadrants 1 and 2 (because in Quadrant 1, both \(x\) and \(y\) are positive, and \(\sin\alpha=\frac{y}{r}\); in Quadrant 2, \(y\) is positive and \(x\) is negative, so \(\sin\alpha=\frac{y}{r}>0\) as \(r>0\) always).
• Cosine is positive (\(\cos\alpha>0\)) in Quadrants 1 and 4 (because \(\cos\alpha = \frac{x}{r}\), so when \(x>0\), which is in Quadrants 1 (\(x>0,y>0\)) and 4 (\(x>0,y<0\))).

Step3: Find the intersection of quadrants

We need the quadrants where both \(\sin\alpha>0\) and \(\cos\alpha>0\). The intersection of the quadrants where \(\sin\alpha>0\) (1,2) and \(\cos\alpha>0\) (1,4) is Quadrant 1.

Answer:

1