Question

Write expressions for other five trigonometric ratios for each acute angle:
a. csc(theta 0) = a/b
b. sec(beta B) = x+1/x-1, x not equal 1

Answer 1

I will do the first one, then you do the second one

cscØ = a/b
or
sinØ = b/a

so make a sketch of a right-angled triangle where the base angle is Ø, its opposite side is b and the hypotenuse is a
let the adjacent side be c
b^2 + c^2 = a^2
c^2 = a^2 - b^2
c = √(a^2 - b^2)

so
sinØ = b/a ---------- cscØ = a/b (that was given)
cosØ = √(a^2-b^2)/a --- secØ = a/√(a^2-b^2)
tanØ = b/√(a^2-b^2) --- cotØ = √(a^2-b^2)/b