Question

If x and y are acute angles, such that cos x + cos y = 3/2 and sin x + sin y = 3/4, then find sin (x + y),
1. 2/5
2. 3/4
3. 3/5
4. 4/5

Answer 1

We have to find the answer by using the trigonometric formulas and identities. First, rewrite both the equations given in question using the appropriate formulas and then divide one equation with another and then, we will get an equation which we will use to find the value of sin (x + y).

Complete step by step answer:
Given, x and y are acute angles.
$cos x + cos y = 3/2$-----(1)
$sin x + sin y = 3/4$------(2)
By using formula $\cos C + \cos D = 2\cos \dfrac{{\left( {C + D} \right)}}{2}.\cos \dfrac{{(C - D)}}{2}$, we can rewrite equation (1) as
$2\cos \dfrac{{\left( {x + y} \right)}}{2}\cos \dfrac{{\left( {x - y} \right)}}{2} = \dfrac{3}{2}$------(3)
By using the formula $\sin C + \sin D = 2\sin \dfrac{{\left( {C + D} \right)}}{2}\cos \dfrac{{\left( {C - D} \right)}}{2}$, we can rewrite equation (2) as
$2\sin \dfrac{{\left( {x + y} \right)}}{2}\cos \dfrac{{\left( {x - y} \right)}}{2} = \dfrac{3}{4}$------(4)
We will divide equation (4) by equation (3) and we will get,
$\dfrac{{\left( {2\sin \dfrac{{\left( {x + y} \right)}}{2}\cos \dfrac{{\left( {x - y} \right)}}{2}} \right)}}{{\left( {2\cos \dfrac{{\left( {x + y} \right)}}{2}\cos \dfrac{{\left( {x - y} \right)}}{2}} \right)}} = \dfrac{{\left( {\dfrac{3}{4}} \right)}}{{\left( {\dfrac{3}{2}} \right)}}$
$\tan \dfrac{{\left( {x + y} \right)}}{2} = \dfrac{1}{2}$------(5)
Now, by using formula $\sin 2A = \dfrac{{2\tan A}}{{1 + {{\tan }^2}A}}$, we will find value of sin (x + y),
$\sin \left( {2\left( {\dfrac{{x + y}}{2}} \right)} \right) = \dfrac{{2\tan \left( {\dfrac{{x + y}}{2}} \right)}}{{1 + {{\tan }^2}\left( {\dfrac{{x + y}}{2}} \right)}}$
By using equation (5), we will put value of $\tan \dfrac{{x + y}}{2}$ in above equation, we get

$\sin (x + y) = \dfrac{{2 \times \dfrac{1}{2}}}{{1 + {{\left( {\dfrac{1}{2}} \right)}^2}}}$
$\sin \left( {x + y} \right) = \dfrac{1}{{\left( {\dfrac{5}{4}} \right)}}$
$\sin \left( {x + y} \right) = \dfrac{4}{5}$

So, the correct answer is “Option 4”.

Note: Trigonometric identities and operations associated with them must be remembered by students to solve these kinds of questions. There are chances that students may make mistakes in calculations so it is advised that they go step by step and not hurry in doing the calculations.