a point on the terminal side of an angle θ in standard position has cos<0 and tanx<0. what quadrant does the terminal side lie in

Answer 1

Because I'm going to be talking about x and y-coordinates, I'm going to use θ in place of x as the name of the angle. (So, assume cos x = cos θ.)

To answer your question, you just need to know that cos θ is the x coordinate of the point corresponding to the angle's position on the unit circle, and sin θ is the y-coordinate of the same point. Also, tan θ=(sin θ)/(cos θ).

The two quadrants on a regular graph (A "Cartesian Plane") where x is negative are the second and third quadrants, to the left of the y-axis. For the tangent of an angle to be negative, the sine and cosine of the angle need opposite signs, since dividing a positive by a negative or vice versa is the way to get a negative answer.

x and y have opposite signs in the second (x is negative, y is positive) and fourth (x is positive, y is negative) quadrants.

Since both the cosine and tangent of an angle are negative when its terminal side is in the second quadrant, that's the answer. QII.

If you're not familiar with how the quadrants work, look here: https://www.purplemath.com/modules/plane3.htm

Have fun with trig!

Answer 2

For Cos x < 0, that means that the projection of the hypotenuse is on the the negative side of the x axis, the possible places for the terminal point for the cosine of the angle to be negative is either in the 2nd or the third quadrant. Now, For tan x < 0, and tan x = sin x /cos x , if cos x is negative then sin x has to be positive and this only occurs in the 2nd quadrant

On the other hand if sin x is negative than cos x has to be positive this can only occur in the 3rd quadrant, but in that quadrant cos x is positive and it has to be -ve. The answer is that the terminal point can only exist in the 2nd quadrant for cos x < 0, and tan x < 0