Question

If cot x= 4/3 and x is not an
angle in the first quadrant, find
the value of sin x + COS X

Answer 1

Answer:

1+cot^2x=cosec^2x=1+(4/3)^2=25/9

cosecx=+5/3 or -5/3

sinx=3/5

cos^2x=1-sin^2x=16/25

cosx=-4/5 or +4/5

Here We will take cosx=-4/5

sinx+cosx=3/5-4/5=-1/5 Ans

Answer 2

Given :  Cot x= 4/3 and x is not an angle in the first quadrant,

To find : the value of sin x + COS X

Solution:

Cot x= 4/3    

Cotx  = Cosx / sinx  

Cotx is + ve  hence

Either cosx and sinx both are positive  or both are negative

in 1st Quadrant cosx and sinx both are positive

and in 3rd Quadrant both are positive

as given that x is not an angle in the first quadrant,  

Hence it must be in 3rd Quadrant  

Hence cosx and sinx must be negative

Cot x= 4/3     = B/P    

H = √B² + P² = 5    ( Using Pythagoras theorem)

Cos x = B/H     =  -4/5      

Sinx = P/H  =  -3/5

sinx + cosx  = -3/5 - 4/5  = -7/5

or use identity

cosec²x = 1 + cot²x

=>  cosec²x = 1 + (4/3)²  = 5²/3²

=> sin²x  = 3²/5²    ( sinx = 1/cosecx)

cos²x + sin²x  = 1  

=> cos²x  = 1 - 3²/5²   = 4²/5²

sin²x  = 3²/5² => sinx = - 3/5  as already shown sign of sinx is negative

cos²x  = 4²/5² => cosx = -4/5   as already shown sign of cosx is negative

sinx + cosx  = -3/5 - 4/5  = -7/5

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