Final answer:
To find the value of 2sinx when tanx equals 7/2 in the third quadrant, we first express sinx and cosx in terms of the given tan value and then apply the Pythagorean identity to determine sinx, which leads to the calculation of 2sinx as -14/sqrt(53).
Step-by-step explanation:
If tanx equals 7/2 and x is in the range of 180 to 270 degrees, we are dealing with an angle in the third quadrant, where both sine and cosine are negative. To find 2sinx, we must first determine the value of sinx. We know that tanx = sinx/cosx, and as given, tanx = 7/2. Using the Pythagorean identity for sine and cosine, namely sin^2x + cos^2x = 1, we can find the other trigonometric functions.
Let cosx be 'b' and sinx be 'a'. Then, we have a/b = 7/2 and since in the third quadrant sine is negative, a = -7 and b = -2. Using the Pythagorean identity, a^2 + b^2 = (-7)^2 + (-2)^2 = 49 + 4 = 53, we can now find sinx as a = sqrt(53).
However, keeping in mind the negative sign that both sine and cosine carry in the third quadrant, sinx must be -7/sqrt(53). Therefore, 2sinx = 2(-7/sqrt(53)), which simplifies to -14/sqrt(53).