let theta be an angle in quadrant 3 so that cos theta= -4/5 find the values of csc theta and tan theta

Question

let theta be an angle in quadrant 3 so that cos theta= -4/5 find the values of csc theta and tan theta

Thankya

Answer 1

cosθ=-4/5

draw a right triangle in quadrant III

cos=adjacent/hypotenuse

this is a 3-4-5 right triangle (the second leg is 3)

sin=opposite/hypotenuse

sinθ=-3/5

csc=1/sin

cscθ=-5/3

tan=opposite/adjacent

tanθ=3/4 (tan is positive in quadrant III)

Answer 2

In quadrant 3, the cosine and sine are negative.

 

We need to find sin(θ)

 

cos²(θ)+sin²(θ) = 1,

 

sin(θ) = -√(1 - cos²(θ))

sin(θ) = -√(1 - (-4/5)²)

sin(θ) = -√(9/25)

sin(θ) = -3/5.

 

Then,

csc(θ) = -5/3,

because it is the reciprocal of sine.

 

 

We use the fact that tan(x) = sin(x)/cos(x)

 

tan(θ) = (-3/5)/(-4/5)

          =3/4.