Question

2 sin inverse 4/5+sin inverse 24/25

Answer 1

SOLUTION

TO DETERMINE

The value of

$\displaystyle \sf{2 { \sin}^{ - 1} \bigg( \frac{4}{5} \bigg) + { \sin}^{ - 1} \bigg( \frac{24}{25} \bigg) }$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on inverse Trigonometric function that

$\displaystyle \sf{2 { \sin}^{ - 1} x = { \sin}^{ - 1} \bigg(2x \sqrt{1 - {x}^{2} } \bigg) }$

EVALUATION

Here the given expression is

$\displaystyle \sf{2 { \sin}^{ - 1} \bigg( \frac{4}{5} \bigg) + { \sin}^{ - 1} \bigg( \frac{24}{25} \bigg) }$

Now

$\displaystyle \sf{2 { \sin}^{ - 1} \bigg( \frac{4}{5} \bigg) }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{4}{5} \times \sqrt{1 - {\bigg( \frac{4}{5} \bigg)}^{2} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{4}{5} \times \sqrt{1 - \frac{16}{25} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{4}{5} \times \sqrt{ \frac{25 - 16}{25} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{4}{5} \times \sqrt{ \frac{9}{25} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{4}{5} \times \frac{3}{5} \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: \frac{24}{25} \: \bigg] }$

Thus we get

$\displaystyle \sf{2 { \sin}^{ - 1} \bigg( \frac{4}{5} \bigg) + { \sin}^{ - 1} \bigg( \frac{24}{25} \bigg) }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg( \frac{24}{25} \bigg) + { \sin}^{ - 1} \bigg( \frac{24}{25} \bigg) }$

$\displaystyle \sf{ =2 { \sin}^{ - 1} \bigg( \frac{24}{25} \bigg) }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{24}{25} \times \sqrt{1 - {\bigg( \frac{24}{25} \bigg)}^{2} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{24}{25} \times \sqrt{1 - \frac{576}{625} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{24}{25} \times \sqrt{ \frac{ 625 - 576}{625} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{24}{25} \times \sqrt{ \frac{ 49}{625} } \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: 2 \times \frac{24}{25} \times \frac{7}{25} \: \bigg] }$

$\displaystyle \sf{ = { \sin}^{ - 1} \bigg[\: \frac{336}{625} \: \bigg] }$

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