if cos x = -1/3 and x is in quad III find the tan x cot x + csc x?

Question

please can you show me the steps.thank you

Answer 1

cos(x) = -1/3
|sin(x)| = √(1-(1/3)²) = √(8/9) = 2√2/3
x is in the third quadrant: sin(x) must be negative.
sin(x) = -2√2/3

cot(x) = 1/tan(x)

tan(x)cot(x) + csc(x)
= tan(x)/tan(x) + 1/sin(x)
= 1 + (-3/2√2)
= 1 - 3√2/4
= (4-3√2)/4

Answer 2

cos x=-1/3
as cos x = adjacent side/ hypotenuse=1/3
so the opppsite side is =sqrt[(3)^2-(1)^2]=sqrt[9-1]=sqrt[8]

so tan x= opposite side/adjacent side =[sqrt(8)/1]
cot x=1/tan x=1/[sqrt(8)/1]=1/sqrt(8)

csc x= hypotenuse/opposite side = -3/sqrt(8)

so
tan x cot x + csc x

substitute the values
[sqrt(8)*1/sqrt(8)] +[ -3/sqrt(8)]
[(sqrt(8) -3)/sqrt(8)] or[1-(3/sqrt(8))]

Answer 3

cosx = - 1 / 3 , third quadrant , x= -1 , r = 3
we have , x ^ 2 + y ^2 = r ^2
y ^2 = r ^2 - x ^2 = ( 3) ^2 - ( - 1) ^2 = 8
y = 2 square root(2) , and , y = - 2 [square root(2)]
in third quadrant , y < 0 , we get y = - 2[ square root (2)]
sinx = y / r = - [ 2square root( 2)] / 3
cscx = 1 / sinx = - 1 /{ [ 2 square root (2)] / 3}
cscx = - 3 / [ 2 square root (2)] =
= - [(3)( square root(2))] / 2 [(square root (2)] ^2
cscx = - [ 3 square root (2) ] / 4

we have ,tanx = 1 / cotx
so
tanx cotx + cscx = (1/ cotx) cotx + cscx = 1 + cscx
tanx cotx + cscx = 1 - {[ 3 square root (2)] / 4 }

Answer 4

tanx cotx + csc x = 1 - 3/(2*2^0.5)

Answer 5

If cos x = -1/3 and x is in the 3rd quadrant, you can easily work out the 3rd side of the triangle using Pythagoras. It is the opposite side, and it is equal to -2√2

So tan x would be -2√2/-1 = 2√2
cot x = 1/ tan x = 1/2√2

Csc x = -3/2√2

Remember that the radius is always positive, the line extending up is positive, down is negative, right is positive and left is negative.

I'll leave the adding and multiplying to you.

Answer 6

k

Answer 7

tan x cot x + cosec x
1 + 1 / sin x
1 + 1 / (- √8 / 3)
1 - 3/√8
(√8 - 3) / √8
(2√2 - 3) / (2√2)

Answer 8

Numerator = sin(x) +tan(x) = sin(x){a million+a million/cos(x)} =sin(x){cos(x) +a million}/cos(x) Denominatoe= sec(x) -cos(x) = a million/cos(x) - cos(x) ={a million-cos^2(x)}/cosx=sin^2(x)/cos(x) Divide N and Denominator cos(x) and sin(x) will cancel LS ={a million+cos(x)/sin(x) = cosec(x) + cot(x) =RS