find acute angles A and B satisfying cot(A+B)=1 , cosec (A-B)=2
To find the acute angles A and B satisfying cot(A + B) = 1 and cosec(A - B) = 2, we can use the trigonometric identities and algebraic manipulations.
Let's start with the first equation:
cot(A + B) = 1
Using the identity cot(x) = 1/tan(x), we can rewrite the equation as:
tan(A + B) = 1
Now, we can find the values of A + B by taking the inverse tangent (tan^(-1)) of both sides:
A + B = tan^(-1)(1)
The inverse tangent of 1 is π/4 (45 degrees). Therefore, we have:
A + B = π/4
Now, let's move on to the second equation:
cosec(A - B) = 2
Using the identity cosec(x) = 1/sin(x), we can rewrite the equation as:
sin(A - B) = 1/2
Now, we can find the values of A - B by taking the inverse sine (sin^(-1)) of both sides:
A - B = sin^(-1)(1/2)
The inverse sine of 1/2 is π/6 (30 degrees). Therefore, we have:
A - B = π/6
Now, we have two equations:
A + B = π/4
A - B = π/6
To solve these equations and find the values of A and B, we can add the two equations together:
(A + B) + (A - B) = π/4 + π/6
Simplifying:
2A = (3π + 2π) / 12
2A = 5π / 12
Dividing both sides by 2:
A = 5π / 24
Next, we can subtract the second equation from the first equation:
(A + B) - (A - B) = π/4 - π/6
Simplifying:
2B = (3π - 2π) / 12
2B = π / 12
Dividing both sides by 2:
B = π / 24
Therefore, the acute angles A and B satisfying cot(A + B) = 1 and cosec(A - B) = 2 are:
A = 5π / 24
B = π / 24
in degrees,
A+B=45
A-B=30
A = 37.5
B = 7.5