cos theta =3/5theta is in not in 1 quadrant, then show that 5sin theta -3 tan theta 3 sec theta - 4 cot theta=
Answers
Answer:
20
Step-by-step explanation:
Given ,
cosθ = 3/5
θ is not in '1st' quadrant
To Find :-
Value of :-
5sinθ - 3tanθ.3secθ - 4cotθ
How To Do :-
As they gave the value cosθ and we can observe that it is positive value . We know that 'cos' ratio is positive only in 1st quadrant and 4th quadrant . As they given that 'θ ' does not belongs to 1st quadrant. → θ belongs to 4th quadrant. So by using the Pythagoras theorem we need to find the value of another side and we need to find the value of 'sinθ , tanθ , secθ , cotθ '.
Formula Required :-
Pythagoras theorem :-
(hypotenuse side)² = (adjacent side)² + (opposite side)²
Trigonometric ratios :-
sinθ = opposite side/hypotenuse side
cosθ = adjacent side/hypotenuse side
tanθ = opposite side/adjacent side
secθ = hypotenuse side/adjacent side
cotθ = adjacent side/opposite side
In 4th quadrant :-
'sin' ratio is negative
'cos' ratio is positive
'tan' ratio is negative
'sec' ratio is positive
'cot' ratio is negative
Solution :-
cosθ = 3/5
adjacent side/hypotenuse side = 3/5
→ adjacent side = 3 , hypotenuse side = 5
Let , opposite side be 'x'
Applying Pythagoras theorem :-
(5)² = (x)² + (3)²
25 = x² + 9
x² = 25 - 9
x² = 16
x = √16
x = 4
∴ Opposite side = x = 4.
sinθ = opposite side/hypotenuse side
= -4/5 [ ∴ 'sin' ratio is negative in 4th quadrant ]
tanθ = opposite side/adjacent side
= -4/3 [ ∴ 'tan' ratio is negative in 4th quadrant ]
secθ = hypotenuse side/adjacent side
= 5/3
cotθ = adjacent side/opposite side
= -3/4 [∴ 'cot' ratio is negative in 4th quadrant ]
5sinθ - 3tanθ .3secθ - 4cotθ
= 5(-3/5) -[ 3(-4/3).3(5/3)] - 4(-3/4)
= -3 -[-4(5)] + 3
= - 3 + 20 + 3
= 20
∴ 5sinθ - 3tanθ .3secθ - 4cotθ = 20.