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given cos x=-3/4, with x in quadrant 2. find the remaining five trigonometric functions of x.

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1 Answer

"x in quadrant 2" means the terminal ray of the angle x is in quadrant 2.
 
If a and b are positive numbers, then (-a,b) is a point in quadrant 2.
 
We know any point can also be represented as (r cos(x), r sin(x) ), so:
 
(r cos(x), r sin(x) ) = (-a,b)
 
sin(x) = b/r
cos(x) = -a/r = -3/4
 
a = 3, r = 4
 
r^2 = a^2 + b^2
 
16 = 9 + b^2
 
b^2 = 16-9 = 7
 
b = √(7)
 
sin(x) = b/r = √(7)/4
cos(x) = -a/r = -3/4
tan(x) = sin(x)/cos(x) = -√(7)/3
cot(x) = 1/tan(x) = -3√(7)/7
sec(x) = 1/cos(x) = -4/3
csc(x) = 1/sin(x) = 4√(7)/7