Question

if \\(\sin(\theta) > 0\\), then \\(\theta\\) is in which quadrant(s)? note: if there are more than one quadrant, please write your answer 1-3, which means quadrant 1 and 3. question 11 if \\(\cos(\theta) > 0\\), then \\(\theta\\) is in which quadrant(s)? note: if there are more than one quadrant, please write your answer 1-3, which means quadrant 1 and 3. question 12 if \\(\tan(\theta) > 0\\), then \\(\theta\\) is in which quadrant(s)? note: if there are more than one quadrant, please write your answer 1-3, which means quadrant 1 and 3.

Explanation:

Question (about \(\sin(\theta) > 0\))

Step1: Recall sine sign rules

In unit circle, \(\sin(\theta)\) is \(y\)-coordinate. In quadrant 1, \(y>0\); quadrant 2, \(y>0\); quadrant 3, \(y<0\); quadrant 4, \(y<0\).

Step2: Determine quadrants

Since \(\sin(\theta)>0\) ( \(y>0\) ), \(\theta\) is in quadrant 1 and 2.

Step1: Recall cosine sign rules

\(\cos(\theta)\) is \(x\)-coordinate. In quadrant 1, \(x>0\); quadrant 2, \(x<0\); quadrant 3, \(x<0\); quadrant 4, \(x>0\).

Step2: Determine quadrants

Since \(\cos(\theta)>0\) ( \(x>0\) ), \(\theta\) is in quadrant 1 and 4.

Step1: Recall tangent definition

\(\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\). For \(\tan(\theta)>0\), \(\sin(\theta)\) and \(\cos(\theta)\) must have same sign.

Step2: Analyze quadrants

• Quadrant 1: \(\sin>0\), \(\cos>0\) (same sign, \(\tan>0\)).
• Quadrant 2: \(\sin>0\), \(\cos<0\) (different sign, \(\tan<0\)).
• Quadrant 3: \(\sin<0\), \(\cos<0\) (same sign, \(\tan>0\)).
• Quadrant 4: \(\sin<0\), \(\cos>0\) (different sign, \(\tan<0\)).

So \(\theta\) is in quadrant 1 and 3.

Answer:

1-2

Question 11 (about \(\cos(\theta) > 0\))