if cos☼=sqrt3/5 and pi<☼<3pi/5 find the other five values (sin tan scs cot csc)?

Answer 1

First of all, pi < x < 3pi/5 doesn't make any sense.

This is how you do it. However, depending on what you actually meant by your constraints, the signs on some of these may be different.

To fix your signs...
Q1 (0 to pi/2) = everything is positive
Q2 (pi/2 through pi) = sin and sec are the only positives, the rest are negative.
Q3 (pi through 3pi/2) = tan and cot are the only positives, the rest are negative.
Q4 (3pi/2 through 2pi) = cos and csc are the only positives, the rest are negative.

Anyway....
Draw a right triangle. The one leg is (sqrt 3), and the hypotenuse is (5). Using pythagorean theorem, the other leg is
sqrt [5^2 - (sqrt 3)^2] = sqrt (25 - 3) = sqrt 22

sin x = (sqrt 22)/5

tan x = (sqrt 22)/(sqrt 3) = sqrt (22/3)

sec x = 1/cos x = 5/(sqrt 3) = (5 sqrt 3)/3

csc x = 1/sinx = 5 / (sqrt 22) = (5 sqrt 22)/ 22

cot x = 1/tan x = sqrt (3/22)

Answer 2

You have some typos: scs is not a trig function. You must mean secant. Usually we specify the quadrant of the angle, so perhaps you mean π < Θ < 3π/2? And is cos Θ = (√3)/5 or is cos Θ = √(3/5) ? I'll assume it's the latter. Then sides of the reference triangle are √2, √3, and √5.

But wait. If π < Θ < 3π/2, then cos should be NEGATIVE. Did you leave that out, too? Assuming you did, then the √3 side of the triangle is on the negative x-axis, the √5 side is the hypotenuse, and the √2 side is below the x-axis in quad 3. Then

sin Θ = -√2 / √5
cos Θ = -√3 / √5
tan Θ = √2 / √3
csc Θ = -√5 / √2
sec Θ = -√5 / √3
cot Θ = √3 / √2

Answer 3

Please allow me to use x instead of the sun...

IDid you mean pi3pi/5 ...

Between pi and 2pi the sine function is negative,
hence,

sin x = -sqrt[1 - cos^2 (x)] = - sqrt { 1 - [sqrt(3)/5]^2} =
-sqrt(1 - 3/5^2) = -sqrt (1 - 3/25) = - sqrt(22/25) = -sqrt(22)/5

tan x = sin x / cos x = [- sqrt(22)/5]/[sqrt(3)/5] = -sqrt(22)/sqrt(3) = - sqrt(66)/3

sec x = 1/cos(x) = 1/(sqrt(3)/5) = 5/sqrt(3) = 5sqrt(3)/3

cot x = 1/tan x = -3/sqrt(66) = -3sqrt(66)/66 = - sqrt(66)/22

csc x = 1/sin x = 1/[- sqrt(22)/5] = -5/sqrt(22) = -5sqrt(22)/22