It is known that arctan(∆) is negative, Which of these statements is/are a correct?

Question

It is known that arctan(x) is negative, Which of these statements is/are a correct?

l.cos(arctan(x)) = negative ll. sin(arctan(x)) = negative

lll.cos(arctan(x)) = positive lll. sin(arctan(x)) = positive

Honestly, I think its between ll. and lll, but I am not sure

Answer 1

Hey Bill,

Since we are told that arctan(x) is negative, and we know that the range for the arctan function is from -pi/2 to pi/2, then we know that the angle we are dealing with is between 0 and -pi/2. This area is known as the fourth quadrant of our unit circle. For angles that are in the fourth quadrant, the cosine function is positive and the sine function is negative.

So your intuition was correct, it is both II and III.

Answer 2

arctan(x) takes any real number and gives you a number between, but not including, -pi/2 and pi/2

arctan(x) is negative, so we know -pi/2 < arctan(x) < 0

lets look at a unit circle. https://en.wikipedia.org/wiki/Unit_circle#/media/File:Unit_circle_angles_color.svg

cos(x) takes in some amount of radians going around the circle starting at (1,0), and tells us the x position of where we land. for positive numbers we go counterclockwise and positive numbers we go clockwise.

we are between 0 (staying at (1,0)) and -pi/2 (rotating down to (0, -1)). so, x must be between 0 and 1.

sin(x) works the same way, but it tells us the y position. so y must be between -1 and 0.

cos(arctan(x)) is positive and sin(arctan(x)) is negative.