Question

$if \: \sin(a + b ) = \frac{ \sqrt{3} }{2} \: and \: sin \: b \: = \frac{1}{2} then \: find \: the \: value \: of \: a \: and \: b$
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Answer 1

Answer

• a = 30°
• b= 30°

Explaination

Given

$\mathtt{ sin(a + b) = \frac{\sqrt{3}}{2}}$

$\mathtt{\implies\: sin(a + b) = sin 60°}$

Cancelling sin from both sides

$\mathtt{\implies\: \cancel{sin}(a + b) = \cancel{sin} 60°}$

$\mathtt{\implies\: a + b = 60° \rightarrow\: (1)}$

and

$\mathtt{\sin b = \frac{1}{2}}$

Cancelling sin from both sides

$\mathtt{\implies\: sin b = sin 30° }$

$\mathtt{\implies\: \cancel{sin} b = \cancel{sin} 30° }$

$\mathtt{\implies\: b = 30° \rightarrow\: (2)}$

From (1) :

a + b = 60°

a + 30° = 60° {putting b = 30°}

a = 60° - 30°

a = 30°

Answer 2

$\huge{ \bold{ \underline{ \purple{ \sf{Detailed \: Answer}}}}}$

• To find : values of a & b
• Given : sin(a+b) =1/2 and

sinb = √3/2

we know sinx is a periodic function ,of period 2π , so it's values repeats after 2π

so let a,b ∈ [ 0,2π]

$\sin(b) = \frac{1}{2}$

↪ b = π/6 .....(1)

$\sin(a + b) = \frac{ \sqrt{3} }{2}$

$\sin(a + b) = \sin( \frac{\pi}{3} )$

thus,

$a + b = \frac{\pi}{3}$

$a = \frac{\pi}{3} - b$

put b = π/6 from ...(1)

$a = \frac{\pi}{3} - \frac{\pi}{6}$

$a = \frac{\pi}{6}$

therefore, a=b =π/6 or 30°