Question

If sin (A + B) = 1 and cos (A - B) = 1, $0\le \left( A+B \right)\le 90$, $A\ge B$, find A and B.

Answer 1

Hint:We know that sin 90˚ = 1 and cos 0˚ = 1. So, replace 1 with sin 90˚ and cos 0˚ and we will try to find the relation between A and B. We will get two relations, (A + B) = 90 and (A – B)= 0. Using these two relations, we will find the value of A and B individually.

Complete step-by-step answer:
It is given in the question that sin (A + B) = 1 and cos (A - B) = 1. It is also given that the value of A + B lies between 0˚ and 90˚, that is, $0\le \left( A+B \right)\le 90$. It is also given that $A\ge B$, then we have to find the value of A and B. We know that sin 90˚ = 1 and cos 0˚ = 1. We will use these two values in the question to find the values of A and B.
We have been given that sin (A + B) = 1. We know that sin 90˚ = 1. So on replacing 1 with sin 90˚ in the first relation, we will get,
sin (A + B) = sin 90˚
We can equate the angles on both sides of the above equation, so we will get,
(A + B) = 90˚………(i)
Since in the question, we have a condition that $0\le \left( A+B \right)\le 90$ and we have got the value of (A + B) = 90, we can say that this condition is satisfied. So, we can proceed further.
Now, we have been also given that,
cos (A - B) = 1
We know that cos 0˚ = 1. So on replacing 1 with cos 0˚ in the second relation, we will get,
cos (A - B) = cos 0˚
We can equate the angles on both sides of the above equation, so we will get,
(A - B) = 0
On transferring –B from the LHS to the RHS, we will get,
A = B………(ii)
Now, our condition was $A\ge B$ and here we have A = B. Since it satisfies the given condition, we can proceed further.
As A and B are equal, we can write equation (i) as,
(A + A) = 90˚
2 A = 90˚
$\text{A}=\dfrac{90~}{2}$
A = 45˚
As we know that, A = B, we will get the value of B also as 45˚.
Therefore, the value of A is 45˚ and the value of B is 45˚.

Note: While solving this question, the students may take the value of cos 90˚ as 1 and the value of sin 0˚ as 1 in a hurry. This is a common silly mistake that the students can make in the exams. So, students will end up formulating two different equations and the value of A and B will also be different. Thus it is recommended to do the calculations precisely.Also students should remember the important standard trigonometric angles and formulas for solving these types of questions.