if cosecø+cotø=1/3, find cosø and determine the quadrant in which ø lies
Answers
Step-by-step explanation:
2x-1/3=1/5-6
6x-3/3=1-5x/5
30x-5=3-15x
45x=8
x=8/45
Step-by-step explanation:
cosecø+cotø=1/3
=> (cosecø+cotø)²=(1/3)²
=> cosec²ø + cot²ø = (1/9)+2cosecø.cotø ---------- (1)
(cosecø+cotø)/cosecø-cotø * (cosecø-cotø)
1/(cosecø-cotø) = 1/3
=> (cosecø-cotø) = 3
=>(cosecø-cotø)² = 3²
=> cosec²ø + cot²ø = 9-2cosecø.cotø -------------- (2)
by (1) and (2),
=> (1/9)+2cosecø.cotø = 9-2cosecø.cotø
4cosecø.cotø = 9-(1/9)
4cosecø.cotø = 80/9
cosecø.cotø = 20/9
9cosø = 20sin²ø
9cosø = 20(1-cos²ø)
20cos²ø+9cosø-20 = 0
let cosø=x
=> 20x²+9x-20 = 0
=> 20x²+25x - 16x -20 = 0
=> 20x²- 16x+25x -20 = 0
=> 4x(5x-4) + 5(5x-4) = 0
=> (5x-4)(4x+5) = 0
=> x = 4/5 or x= -5/4
when x = 4/5
ie., when cosø = 4/5
=> 0 < cosø < 1
=> 90° < ø < 0°
=> ø will not lie in the First quadrant
when x = -5/4
ie., when cosø = -5/4
=> 0 > cosø < -1
=> 90° > ø < 180°