If θ is an angle in standard position and tan θ = −7/2, in which quadrant can θ be located?
a. quadrant 3 and quadrant 4
b. quadrant 2 and quadrant 4
c. quadrant 2 and quadrant 3
d. quadrant 1 and quadrant 2
help
tanθ = y/x
so, you need x and y to have different signs...
so is it a, b, c, or d?
To determine the quadrant where θ can be located based on the given information, we need to consider the trigonometric functions in each quadrant.
The tangent function (tan) is negative only in the second and fourth quadrants.
To confirm this, we can use the definition of tangent as the ratio of sine (opposite side) to cosine (adjacent side). In the second and fourth quadrants, the sine is positive, while the cosine is negative, making the tangent negative.
Thus, we can eliminate options a and c, which include a quadrant other than the second or fourth.
Now, to further narrow down our answer, let's consider the specific value of tan θ, which is -7/2.
Given that tan θ = -7/2, we know that the ratio of the opposite side to the adjacent side of the angle θ is -7/2.
In the second quadrant, the opposite side is positive, while the adjacent side is negative. Hence, the ratio is positive.
However, in the fourth quadrant, both the opposite side and the adjacent side are negative. Therefore, the ratio is negative.
Since we are given that tan θ = -7/2, the only quadrant where this can be true is the fourth quadrant.
So, the correct answer is option b: quadrant 2 and quadrant 4.