Question

In which quadrant does ø lie if the following statements are true

sec ø < 0 and csc ø > 0

Answer 1

Ф lies in the 2nd quadrant

Explanation:

1. If angle Ф lies in the 1st quadrant, then

• sin Ф is a positive value

• cos Ф is a positive value

• tan Ф is a positive value

2. If angle Ф lies in the 2nd quadrant, then

• sin Ф is a positive value

• cos Ф is a negative value

• tan Ф is a negative value

3.If angle Ф lies in the 3rd quadrant, then

• sin Ф is a negative value

• cos Ф is a negative value

• tan Ф is a positive value

4. If angle Ф lies in the 4th quadrant, then

• sin Ф is a negative value

• cos Ф is a positive value

• tan Ф is a negative value

∵ sec Ф < 0

∴ sec Ф is a negative value

∵ cos Ф = 1/sec Ф

∴ cos Ф is a negative value

∵ csc Ф > 0

∴ csc Ф is a positive value

∵ sin Ф = 1/csc Ф

∴ sin Ф is a positive value

→ By using the second rule above

∴ Ф lies in the 2nd quadrant