Question

Prove that :-
$\sin12 \degree \: \sin48 \degree \: \sin54 \degree \: = \dfrac{1}{8}$
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Answer 1

Step-by-step explanation:

Given :-

Sin 12° Sin 48° Sin 54°

Required To Prove :-

Prove that Sin 12° Sin 48° Sin 54° = 1/8

Solution :-

On taking LHS :

Sin 12° Sin 48° Sin 54°

On multiplying and dividing by 2 then

=> (1/2)[2 Sin 12° Sin 48° Sin 54°]

It can be rearranged as

=> (1/2) [2 Sin 12° Sin 48°] Sin 54°

=> (1/2) [ 2 Sin 48° Sin 12° ] Sin 54°

We know that

Sin A Sin B = (1/2)[Cos (A-B) - Cos(A+B)]

We have , A = 48° and B = 12°

=> (1/2)[2(1/2){Cos(48°-12°)-Cos(48°+12°)}] Sin 54°

=> (1/2)[ (2/2) Cos 36° - Cos 60° ] Sin 54°

=> (1/2) [ (Cos 36° - Cos 60° ] Sin 54°

We know that

Cos 60° = 1/2

=> (1/2) [ Cos 36° - (1/2) ] Sin 54°

=> (1/2) [ (2 Cos 36°-1)/2 ] Sin 54°

=> (1/4) [ 2 Cos 36° -1] Sin 54°

=> (1/4) [ 2 Cos 36° Sin 54° - Sin 54° ]

We know that

Cos A Sin B = (1/2) [ Sin (A+B) - Sin (A-B) ]

We have , A = 36° , B = 54°

=> (1/4) [ 2 (1/2) Sin (36°+54°) -Sin (36-54°) -Sin 54°]

=> (1/4) [(2/2) Sin 90° - Sin (-18° ) - Sin 54°]

We know that

Sin (-A) = -Sin A

=> (1/4) [ Sin 90° + Sin 18° - Sin 54° ]

We know that

Sin 90° = 1

=> (1/4) [ 1 + Sin 18° - Sin 54° ]

We know that

Sin 18° = (√5-1)/4

Sin 54° = (√5+1)/4

On Substituting these values in the above expression

=> (1/4) [ 1+ { (√5-1)/4} - {(√5+1)/4}]

=>(1/4) [ 1+ {(√5-1)-(√5+1)}/4]

=> (1/4) [ 1+ {(√5-1-√5-1)/4} ]

=> (1/4) [ 1+ {(-1-1)/4} ]

=> (1/4) [1+(-2/4) ]

=> (1/4) [ 1+(-1/2) ]

=> (1/4) [ (2-1)/2)]

=> (1/4) (1/2)

=> (1×1)/(2×4)

=> 1/8

=> RHS

=> LHS = RHS

Hence, Proved.

Answer:-

Sin 12° Sin 48° Sin 54° = 1/8

Used formulae:-

→ Sin A Sin B = (1/2)[Cos (A-B) - Cos(A+B)]

→ Cos A Sin B = (1/2) [ Sin (A+B) - Sin (A-B) ]

→ Sin (-A) = -Sin A

→ Sin 18° = (√5-1)/4

→ Sin 54° = (√5+1)/4

→ Sin 90° = 1

→ Cos 60° = 1/2

Answer 2

Answer:

Sonali and Rupali are partners in a firm with capital of 600000 each. They decided to admit Rakhi as a partner with 1/4th share in the profits and losses of the firm. Rakhi bring of 800000 as her share of capital. Their profit and loss account shows a credit balance of 400000 as on the date of admission. Give journal entry to record Goodwill on rakhi's admission

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