If TanTheta= -65/72 and the terminal side of theta lies in Quadrant II use a pythagorean identity to find sec theta.
sec^2 = 1+tan^2, so
sec^2 = 1+(65/72)^2
In QII, cos and sec are negative
To find sec(theta) using a Pythagorean identity, we need to determine the value of cos(theta).
Given that the terminal side of theta lies in Quadrant II and tan(theta) = -65/72, we can use the fact that tangent is negative in Quadrant II to determine that the opposite side is -65 and the adjacent side is 72.
Using the Pythagorean theorem, we can find the hypotenuse of the triangle formed by these sides:
h² = (-65)² + 72²
h² = 4225 + 5184
h² = 9409
Taking the square root of both sides, we find:
h = √9409
h ≈ 97
Now that we know the lengths of the sides, we can find cos(theta) using the adjacent side and the hypotenuse:
cos(theta) = adjacent/hypotenuse
cos(theta) = 72/97
Finally, to find sec(theta), we can use the reciprocal identity of cosine:
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (72/97)
sec(theta) = 97/72
Therefore, sec(theta) is equal to 97/72.