What are sin x, tan x, cot x, if 180<x<270 and cos x=-4/5?
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We have cos x = -4/5. Also 180< x< 270. For these values of x sin x is negative and so is cos x.
We use the relation (cos x)^2 + (sin x)^2 = 1
=> (-4/5)^2 + (sin x)^2 = 1
=> (sin x)^2 = 1 - (16 / 25)
=> (sin x)^2 = 9/25
=> sin x = -3/5
tan x = sin x / cos x = (-3/5)/(-4/5) = 3/4
cot x = cos x / sin x = 4/3
sin x = -3/5 , tan x = 3/4 and cot x = 4/3.
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The angle x is in the 3rd quadrant, so the value of the function sine is negative, but the values of teh functions tangent and cotangent are positive.
We'll apply the fundamental formula of trigonometry:
(sin x)^2 + (cos x)^2 =1
(sin x)^2 = 1 - (cos x)^2
(sin x)^2 = 1 - (-4/5)^2
(sin x)^2 = 1 - 16/25
(sin x)^2 = 9/25
sin x = -3/5
tan x = sin x/cos x
tan x = -3/5/-4/5
sin x = -3/5, tan x = 3/4 and cot x = 4/3
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