If cos ⁡x=−1/2, and 90°<x<180°, find cos x/sin x

Question

Please I need a Step by step procedure of this calculation accordingly, it's urgent

Simply stated and not too complicated

Thanks

Answer 1

We are in second quadrant, so sinx will be positive

Fundamental trig identity:

cosx^2 + sinx^2 = 1

(-1/2)^2 + sinx^2 = 1

sinx^2 = 3/4

sinx = sqrt(3)/2

cosx/sinx = -1/2

-----

sqrt(3)/2

= -1/sqrt(3) = -sqrt(3)/3

Answer 2

Drawing and labeling an appropriate triangle,

sin x = √3 / 2

cos x / sin x = (-1/2) / (√3 / 2)

Answer 3

cosx = -1/2 in quadrant II

reference angle is formed by a 30-60-90 right triangle

a reference angle of 60 degrees in the 2nd quadrant is equivalent to 120 degrees

but if you don't see that, just use a calculator with an inverse cosine function. Punch in -0.5 and it will say 120 degrees or

its set for radians, it will say 2pi/3

x = 120 degrees or 2pi/3 radians

sin120 = sqr3/2 = about 1.732/2 = 0.866

sinx = about 0.866 or = exactly (1/2)sqr3

or use a calculator with a sine function

plug in 120 and out pops 0.866025403. round it off

cosx/sinx = (-1/2)/0.866 = -0.5/0.866 = -0.577

just use a calculator and divide .5 by .866 to get .577, and stick a negative sign in front of it.

or round it off to -0.58 or to -0.6 = -3/5

or

cosx/sinx = (-1/2)/(1/2)sqr3 = (-1/2)(2/sqr3) = -1/sqr3 =-sqr3/3

cosx/sinx = -(1/3)sqr3 = about (-1/3)(1.732) = -1.732/3 = -0.577

or

cosx/sinx = cotx = adjacent side/opposite side = (-1/2)/sqr3/2

= -.5/(1.732/2) = -.5/.867 = -.577

or

cosx/sinx = (-1/2)(2/sqr3)= -1/sqr3= -sqr3/3 = -1.732/3 = -0.577

some calculators only have tangent functions, and no cotangent

if that's what you have, find the tan120= -1.73205

then invert it -1/1.73205 = -0.577