Question

Identify the quadrant for the angle ⊖ satisfying the following conditions.

1) tan ⊖ > 0 and sin ⊖ < 0.
2) sec ⊖ < 0 and tan ⊖ < 0.
3) tan ⊖ < 0 and sin ⊖ < 0

Answer 1

To satisfy the given conditions, the angle ⊖ is in the third quadrant (Q3) for condition 1, in the second quadrant (Q2) for condition 2, and in the fourth quadrant (Q4) for condition 3.

Step-by-step explanation:

To identify the quadrant for a given angle, we need to analyze the signs of the trigonometric functions involved. Let's go through each condition:

1. Condition 1: tan ⊖ > 0 and sin ⊖ < 0. In this case, tan ⊖ is positive and sin ⊖ is negative. The only quadrant where both tan and sin are positive is the third quadrant (Q3).
2. Condition 2: sec ⊖ < 0 and tan ⊖ < 0. Since secant is the reciprocal of cosine, when sec ⊖ is negative, cosine must also be negative. So, tan ⊖ can be negative in either Q2 or Q4. However, since we already know sec ⊖ is negative, it means ⊖ is in the second quadrant (Q2).
3. Condition 3: tan ⊖ < 0 and sin ⊖ < 0. Here, both tan ⊖ and sin ⊖ are negative. The only quadrant where both tan and sin are negative is the fourth quadrant (Q4).

Therefore, to satisfy the given conditions:

1. ⊖ is in the third quadrant (Q3).
2. ⊖ is in the second quadrant (Q2).
3. ⊖ is in the fourth quadrant (Q4).