Question

In which quadrant does θ lie if the following statements are true:

sin
⁡
θ
>
0
and
tan
⁡
θ
>
0
sinθ>0 and tanθ>0

Answer 1

For both sinθ and tanθ to be greater than zero, the angle θ must lie in the first quadrant, where both opposite and adjacent sides of a right triangle are positive.

Step-by-step explanation:

When considering the conditions where both sinθ and tanθ are greater than zero, we must identify in which of the four quadrants both of these trigonometric functions are positive. The sine function represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle, while the tangent function represents the ratio of the length of the opposite side to the adjacent side.

The only quadrant where both sine and tangent are positive is the first quadrant, where all trigonometric functions are positive. This is due to the coordinates of points in the first quadrant being both positive, leading to positive values for all ratios derived from the sides of a right-angled triangle.

Therefore, with both sinθ > 0 and tanθ > 0, the angle θ must lie in the first quadrant.

Answer 2

Quadrant I

Step-by-step explanation:

We are told that sin is positive (greater than 0) and tan is positive (greater than 0).

We know that sin is positive in quadrants I and II.

We also know that tan is positive in quadrants I and III.

Since both conditions share Quadrant I, the answer is Quadrant I.