Question 1011508
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sin^-1 (3÷5) + sin^-1 (5÷13) = sin^-1 (56÷65)
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Thus we need to prove that  arcsin{{{(3/5)}}} + arcsin{{{(5/13)}}} = arcsin{{{(56/65)}}}.

<U>Proof</U>

Let {{{alpha}}} = arcsin{{{(3/5)}}} and {{{beta}}} = arcsin{{{(5/13)}}}.

Thus {{{sin(alpha)}}} = {{{3/5}}} and {{{alpha}}} is in Quadrant 1;   {{{sin(beta)}}} = {{{5/13}}} and {{{beta}}} is in Quadrant 1 also.

We need to prove that {{{sin(alpha + beta)}}} = {{{56/65}}}  and  {{{alpha}}} + {{{beta}}} is in Quadrant 1.

Since {{{sin(alpha)}}} = {{{3/5}}}, you have {{{cos(alpha)}}} = {{{sqrt(1 - (3/5)^2)}}} = {{{4/5}}}        (make all intermediate calculations yourself)

Since {{{sin(beta)}}} = {{{5/13}}}, you have {{{cos(beta)}}} = {{{sqrt(1 - (5/13)^2)}}} = {{{12/13}}}    (make all intermediate calculations yourself)

Now apply the formula for {{{sin(alpha+beta)}}}:

{{{sin(alpha + beta)}}} = {{{sin(alpha)*cos(beta) + cos(alpha)*sin(beta)}}} = {{{(3/5)*(12/13) + (4/5)*(5/13)}}} = {{{(36+20)/(5*13)}}} = {{{56/65}}}.   

Thus the first half of the statement is proved.

Next calculate {{{cos(alpha+beta)}}}:

{{{cos(alpha + beta)}}} = {{{cos(alpha)*cos(beta) - sin(alpha)*sin(beta)}}} = {{{(4/5)*(12/13) - (3/5)*(5/13)}}} = {{{(48-15)/(5*13)}}} = {{{33/65}}}.

Since {{{cos(alpha + beta)}}} > 0, {{{alpha + beta}}} is in the quadrant 1.

The proof is completed.
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