Since cot x = cos x / sin x, if cot x = 1/2, with the angle x in the first quadrant,
then cos x = 1 and sin x = 2
(1) State true or false. Is this a possible situation?
(2) If false, explain why.
the values for sine and cosine are limited to the range ±1
cot x = 1/2 can be written in an infinite number of ways, the 1/2 is simply the ratio expressed in simplest form
how about cot x = .5/1
or how about cotx = .378/.756
etc
by definition cotx = adjacent side/opposite side
in a right-angled triangle
so in the original given cotx = 1/2
the adjacent is 1, and the opposite is 2,
or x = 1 and y = 2
r^2 = 1^2 + 2^1 = 5
r = √5
then sinØ = 2/√5 and cosØ = 1/√5
then tanØ = (2/√5) / (1/√5) = 2/1
and cotØ = 1/2
(1) False. This is not a possible situation.
(2) The equation cot x = cos x / sin x can be rewritten as cos x = cot x * sin x.
If cot x = 1/2, and sin x = 2 (which is greater than 1), the resulting cosine value would be greater than 1 as well, which is not possible since cosine values are always between -1 and 1. Therefore, this situation is not possible.
(1) False. This is not a possible situation.
(2) The reason why this is not possible is that in the first quadrant, both the cosine (cos x) and sine (sin x) of an angle are always positive. Since cot x = cos x / sin x, if cot x = 1/2, then both cos x and sin x would have to be positive. However, if cos x = 1, then sin x would have to be greater than 1 in order for cot x = cos x / sin x = 1/2. Since sin x cannot be greater than 1 in the first quadrant, this situation is not possible.