sinA = 3/5 and cosB =9/41 and A and B lie in the first quadrant, find sin (A+B) .
Answers
Answer:
187/205
Step-by-step explanation:
Given,
sinA = 3/4
cosB = 9/41
A , B lie in first quadrant.
To Find :-
Value of 'sin(A + B)'
How To Do :-
As they gave that A , B lie in the first quadrant i.e in first quadrant all ratios are positive . To find the value of 'cosA and sinB' we need to find the remaining side of them by using pythagoreas theorem and by using the sin(A+B) formula we will get the value of that.
Formula Required :-
Pythogoreas theorem :-
(Hypotenuse side)² = (opposite side)² + (adjacent side)²
sinα = opposite side/hypotenuse side
cosα = adjacent side/hypotenuse side
sin(A + B) = sinAcosB + cosAsinB
Solution :-
sinA = 3/5
→ opposite side/hypotenuse side = 3/5
opposite side = 3 , hypotenuse side = 5
Let,
Adjacent side be 'x'
Applying pythogoreas theorem :-
(5)² = 3² + x²
25 = 9 + x²
25 - 9 = x²
16 = x²
x = √16
x = 4
∴ Adjacent side = x = 4
cosA = adjacent side/hypotenuse side
= 4/5
∴ cosA = 4/5
cosB = 9/41
adjacent side/hypotenuse side = 9/41
adjacent side = 9 , hypotenuse side = 41
Let, opposite side be 'x'
Applying pythagoreas theorem : -
(41)² = 9² + x²
1681 = 81 + x²
1681 - 81 = x²
x² = 1600
x =√1600
x = 40
∴ Opposite side = x = 40
sinB = opposite side/hypotenuse side
∴sinB = 40/41
sin(A + B) = sinAcosB + sinBcosA
= 3/5 × 9/41 + 4/5 × 40/41
= 27/205 + 160/205
= (27 + 160)/205
= 187/205
∴ sin(A + B) = 187/205