Question

Given sinθ=- 3/5 and cscθ=-5/3 in quadrant III, find the value of other trigonometric functions using a Pythagorean Identity. Show your work.

Find the value of cosθ, secθ and tanθ.

Answer 1

Let ABC be a triangle in the 3rd quadrant, right-angled at B.
So, AB-> Perpendicular BC -> Base AC -> Hypotenuse.
Given: sinθ=-3/5 cosecθ=-5/3
According to Pythagorean theorem, square of the hypotenuse is equal to the sum of square of the other two sides.
Therefore in triangle ABC, 〖AC〗^2=〖AB〗^2+〖BC〗^2 ------
--(1)
Since sinθ=Perpendicular/Hypotenuse ,
AC=5 and AB=3
Substituting these values in equation (1)

〖BC〗^2=〖AC〗^2-〖AB〗^2

〖BC〗^2=5^2-3^2


〖BC〗^2=25-9

〖BC〗^2=16

BC=4 units

Since the triangle is in the 3rd quadrant, all trigonometric ratios, except tan
and cot are negative.
So,cosθ=Base/Hypotenuse Cosθ=-4/5
secθ=Hypotnuse/Base secθ=-5/4
tanθ=Perpendicular/Base tanθ=3/4
cotθ=Base/Perpendicular cotθ=4/3