Question

sin(A+B)=K and sin(A-B)=1/2;0<A+B<90°,<A><B.find angleA and angle B.​

Answer 1

Correct Question:-

If sin(A + B) = 1 and sin (A-B) = 1/2, 0 ≤ A + B ≤ 90o and A > B, then find ∠A and ∠B.

Answer:-

Given:

sin (A + B)° = 1

1 can be written as sin 90°.

So,

⟹ sin (A + B)° = sin 90°

On comparing both sides we get,

⟹ ∠A + ∠B = 90° -- equation (1)

Similarly,

⟹ sin (A - B)° = 1/2

⟹ sin (A - B)° = sin 30°

⟹ ∠A - ∠B = 30° -- equation (2)

Add equations (1) & (2).

⟹ ∠A + ∠B + ∠A - ∠B = 90° + 30°

⟹ 2∠A = 120°

⟹ ∠A = 120°/2

⟹ ∠A = 60°

Substitute the value of ∠A in equation (1)

⟹ ∠A + ∠B = 90°

⟹ 60° + ∠B = 90°

⟹ ∠B = 90° - 60°

⟹ ∠B = 30°

Answer 2

$\large{ \underline{ \green{ \sf \: Given :-}}}$

$\rm :\implies\:sin(A + B) = 1$

$\rm :\implies\:sin(A - B) = \dfrac{1}{2}$

$\begin{gathered}\begin{gathered}\bf \: To \: find \:  - \begin{cases} &\sf{angle \: A \: and \: angle \: B}  \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

It is given that

$\rm :\implies\:sin(A + B) = 1$

$\rm :\implies\:sin(A + B) = sin90 \degree$

$\rm :\implies\: \boxed{ \pink{ \bf \: A + B\: = \tt \: 90 \degree \: }} - - - (1)$

Further,

It is given that

$\rm :\implies\:sin(A - B) = \dfrac{1}{2}$

$\rm :\implies\:sin(A - B) = sin30\degree \:$

$\rm :\implies\: \boxed{ \pink{ \bf \:A - B \: = \tt \:30\degree \: }} - - - (2)$

Now,

Adding equation (1) and (2), we get

$\rm :\implies\:2A = 90\degree \: + 30\degree \:$

$\rm :\implies\:2A = 120\degree \:$

$\rm :\implies\: \boxed{ \pink{ \bf \: A \: = \tt \: 60\degree \:}}$

On substituting, A = 60°, in equation (1), we get

$\rm :\implies\:60\degree \: + B = 90\degree \:$

$\rm :\implies\:B = 90\degree \: - 60\degree \:$

$\rm :\implies\: \boxed{ \pink{ \bf \: B \: = \tt \: 30\degree \:}}$

Additional Information

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$