If cot theta = -12/5 and the terminal side of theta lies in quadrant II find secant theta
figure the sides of the right triangle: hypotenuse is 13
secant -13/12
To find secant theta, we first need to find the value of cosine theta since secant theta is the reciprocal of cosine theta.
Given that cot theta = -12/5, we can use the relationship between cotangent and tangent as follows:
cot theta = adjacent side / opposite side
From the given information, we have cot theta = -12/5. Since the terminal side of theta lies in quadrant II, we can conclude that the adjacent side is negative and the opposite side is positive.
Let's denote the adjacent side as -12 and the opposite side as 5. This allows us to set up the equation:
cot theta = adjacent side / opposite side
-12/5 = -12/5
Now, we can use the Pythagorean theorem to find the hypotenuse. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs (adjacent and opposite sides) is equal to the square of the length of the hypotenuse.
Using the values we have, let's calculate the hypotenuse:
adjacent side = -12
opposite side = 5
According to the Pythagorean theorem:
(-12)^2 + (5)^2 = hypotenuse^2
144 + 25 = hypotenuse^2
169 = hypotenuse^2
Taking the square root of both sides, we find:
hypotenuse = √169
hypotenuse = 13
Now that we know the values of the adjacent side, opposite side, and the hypotenuse, we can find cosine theta using the adjacent side and hypotenuse:
cos theta = adjacent side / hypotenuse
cos theta = -12/13
Finally, the secant theta, which is the reciprocal of cosine theta, is:
sec theta = 1 / cos theta
sec theta = 1 / (-12/13)
sec theta = -13/12
Therefore, the value of secant theta is -13/12.