Question

In which quadrant does 0 lie if the following statements are true: tan θ> 0 and sin θ < 0 O Quadrant I O Quadrant III O Quadrant II O Quadrant IV

Answer 1

The angle θ lies in Quadrant IV, as this is the only quadrant where tan θ is positive and sin θ is negative.

Step-by-step explanation:

To determine in which quadrant the angle θ lies given that tan θ > 0 and sin θ < 0, we need to consider the signs of the trigonometric functions in each quadrant:

• In Quadrant I, both sine and tangent are positive.
• In Quadrant II, sine is positive and tangent is negative.
• In Quadrant III, both sine and tangent are negative.
• In Quadrant IV, sine is negative and tangent is positive.

Since we know that tan θ is positive and sin θ is negative, the angle θ must lie in Quadrant IV, where the sine function is negative and the tangent function is positive.

Answer 2

Angle θ lies in the fourth quadrant where tan θ is positive and sin θ is negative as per the trigonometric sign rules.

Step-by-step explanation:

The student asked in which quadrant angle θ lies given that tan θ > 0 and sin θ < 0. The trigonometric sign rule states that tangent is positive where sine and cosine have the same sign, which happens in the first and third quadrants. Since sine is negative, it must be in a quadrant where y-values (sine values) are negative, which occurs in the third and fourth quadrants. Therefore, knowing that both tangent is positive and sine is negative, angle θ must lie in the fourth quadrant.