Question

sin (A+B) = 1, & sin (A-B) = 1/2 ; <A=? <B=?

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Answer 1

Answer :-

sin (A + B) = 1 = sin 90⁰ A + B = 90⁰  ____(i)

sin (A-B) = 1/2 = sin 30⁰ A - B = 30⁰  ____(ii)

Solving eq. (i) and (ii), A = 60⁰ and B = 30⁰

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Answer 2

Given :-

$\quad \leadsto \quad \sf \sin ( A + B ) = 1$

$\quad \leadsto \quad \sf \sin ( A - B ) = \dfrac{1}{2}$

To Find :-

Value of $\sf \angle A \:\: and \:\: \angle B$

Solution :-

Lets use the first given Trigonometric equation as follows ;

$\quad \leadsto \quad \sf \sin ( A + B ) = 1$

We knows that ;

$\quad \qquad { \bigstar { \underline { \boxed { \red { \underbrace { \bf { \sin \dfrac{\pi}{2} = \sin 90° = 1 }}}}}}}{\bigstar}$

Using this we can write the above as ;

${ : \implies \quad \sf \sin ( A + B ) = \sin 90°}$

${ : \implies \quad \sf \sin^{-1} \{\sin ( A + B ) \} = 90}$

${ : \implies \quad \sf \cancel{\sin^{-1}} \{\cancel{\sin} ( A + B ) \} = 90}$

${ : \implies \quad \sf A + B = 90 \quad ----(i) }$

Now , Consider the 2nd given Trigonometric Equation ;

$\quad \leadsto \quad \sf \sin ( A - B ) = \dfrac{1}{2}$

We knows that ;

$\quad \qquad { \bigstar { \underline { \boxed { \red { \underbrace { \bf { \sin \dfrac{\pi}{6} = \sin 30° = \dfrac{1}{2} }}}}}}}{\bigstar}$

Using this we can write the above as ;

${ : \implies \quad \sf \sin ( A - B ) = \sin 30°}$

${ : \implies \quad \sf \sin^{-1} \{\sin ( A - B ) \} = 30}$

${ : \implies \quad \sf \cancel{\sin^{-1}} \{\cancel{\sin} ( A - B ) \} = 30}$

${ : \implies \quad \sf A - B = 30 \quad ----(ii) }$

By elimination method , adding (i) & (ii) we have ;

${ : \implies \quad \sf A - B + A + B = 30° + 90° }$

${ : \implies \quad \sf A - \cancel{B} + A + \cancel{B} = 120 }$

${ : \implies \quad \sf 2A = 120}$

${ : \implies \quad \sf A = \dfrac{120}{2} }$

${ : \implies \quad \therefore \sf \angle A = 60° }$

Also , By eq. (ii) we have ;

${ : \implies \quad \sf 60 - B = 30° }$

${ : \implies \quad \sf 60 - 30 = B }$

${ : \implies \quad \therefore \sf \angle B = 30° }$

Henceforth , The Required Measures of $\sf \angle A$ and $\sf \angle B$ are 60° and 30°