What is the Sum of Angles a and b in This Trigonometry Problem?

[Helly123]

Homework Statement

$\sin a + \cos b$ = $\frac{-1}{2}$
$\cos a + \sin b$ = $\frac{\sqrt 3}{2}$

0 < a < $\pi/2$
$\pi/2$ < b < $\pi$

a + b = ? By calculating sin (a+b)

Homework Equations

The Attempt at a Solution

I tried :
$\sin a + \cos b = 2sin\frac{(a+b)}{2}cos\frac{(a-b)}{2} = -\frac{1}{2}$

$\cos a + \sin b =2sin\frac{(a+b)}{2}cos\frac{(b-a)}{2} = \frac{\sqrt3}{2}$

I tried to multiple it by $\sqrt2/2$
$\sin a \cos 45 + \sin 45 \cos b = -\frac{1}{2}\frac{\sqrt 2}{2}$
$\sin 45 \cos a + \sin b \cos 45 = \frac{\sqrt3}{2}\frac{\sqrt 2}{2}$

$\sin a + \cos b = -\frac{1}{2} = \sin 210 = \sin 330$

None of this steps get me a clue to find a + b. Can i get a clue?

 

[Dr Dr news]

What is your usual approach when solving two simultaneous equations? Don't you try to eliminate one variable?

 

[Wrichik Basu]

Square the two equations and add.

You'll get $\sin a \cos b + \cos a \sin b$ on the left hand side along with other terms. Other terms will reduce to 1. $\sin a \cos b + \cos a \sin b = \sin (a+b)$. You're done.

 

[Chandra Prayaga]

Work at it backwards. The problem asks you to find a + b by first finding sin(a + b). What is sin(a + b) ? Then see if you can get that from the given equations

 

[Helly123]

Square the two equations and add.

You'll get $\sin a \cos b + \cos a \sin b$ on the left hand side along with other terms. Other terms will reduce to 1. $\sin a \cos b + \cos a \sin b = \sin (a+b)$. You're done.

It worked. Thanks