if tan A=3, tan B=1, A and B are acute angle. Find cos(A-B)
Answers
Answer:
( 1 + √3)/2√2
Step-by-step explanation:
Correct Question :-
if tan A=√3, tan B=1, A and B are acute angle. Find cos(A-B)
Given,
tanA = √3 , tanB = 1
A , B are acute angles.
To Find :-
Value of cos(A - B)
How To Do :-
As they given the values of 'tanA' and 'tanB' we need to find the values of 'angle A and angle B' from that. After obtaining those values we need to find the values of 'cosA , cosB , sinA , sinB' and we need to substitute these values in the formula 'cos(A - B)'.
Formula Required :-
tan60° = √3
tan45° = 1
cos(A - B) = cosA.cosB + sinA.sinB
sin60° = √3/2
sin45° = 1/√2
cos60° = 1/2
cos45° = 1/√2
Solution :-
tanA = √3
tanA = tan60°
Cancelling 'tan' on both sides :-
A = 60°
tanB = 1
tanB = tan45°
Cancelling 'tan' on both sides :-
B = 45°
→ cosA = cos60°
= 1/2
cosB = cos45°
= 1/√2
sinA = sin60°
= √3/2
sinB = sin45°
= 1/√2
cos(A - B) = cosA.cosB + sinA.sinB
=( 1/2 × 1/√2) + (√3/2 × 1/√2)
= 1/2√2 + √3/2√2
=( 1 + √3)/2√2
∴ cos(A - B) = ( 1 + √3)/2√2