If sin(x) = -1/2, then find all possible values of sec(x).

 May 25, 2020

sin = -1/2      cos = + or -  sqrt 3/2       sec = 1/cos    =   2 / sqrt 3    or   - 2/ sqrt3

 May 25, 2020

ElectricPavlov beat me to it, but I guess I will present this since it took me half an hour to write.


By definition, the sine function is the ratio of the opposite side of the reference angle to the hypotenuse of a triangle. 

By definition, the secant function is the ratio of the hypotenuse of a triangle to its adjacent side of the reference angle.


First, I would pay attention to the sign of the answer to \(\sin(x)\). As it is right now, it is negative, and there are two quadrants where the sine of an angle is negative: quadrant III and quadrant IV. 


Since the sine function is negative in these two quadrants, I would draw a right triangle with the given ratio of the sides and solve for the third side so that I could generate the ratio of the hypotenuse to the adjacent side of the triangle. I made a diagram so that you can follow along:


Just by inspection, the ratio of this triangle's side lengths is exactly that of a 30-60-90 triangle. The length of the third side, therefore, is \(\sqrt{3}\).


Therefore, \(\sec x=\frac{2}{-\sqrt{3}}=\frac{-2\sqrt{3}}{3}\\ \sec x=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\).

 May 25, 2020

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