Question

if 3sinβ+5cosβ=5,then find the value of 3cosβ-5sinβ​

Answer 1

$3 \sin( \beta ) + 5 \cos( \beta ) = 5 \: \: \: \: \: \: \: \: ...(1)\\ \\ \\ let \: \: \: 3 \cos( \beta ) - 5 \sin( \beta ) = a \: \: \: \: \: \: \: ...(2) \\ \\ \\ squaring \: bothe \: the \: equations \: \\ \\ \\ 9 { \sin}^{2} \beta + 25 { \cos}^{2} \beta + 30 \sin( \beta ) \cos( \beta ) = 25 \: \: \: \: \: \: \: \: ...(3) \\ \\ \\ 9 { \cos }^{2} \beta + 25 { \sin }^{2} \beta - 30 \sin( \beta ) \cos( \beta ) = {a}^{2} \: \: \: \: \: \: \: ...(4)\\ \\ \\ adding \: (3) \: an \: (4) \\ \\ \\ 9( { \sin \beta }^{2} + { \cos }^{2} \beta ) + 25( { \sin }^{2} \beta + { \cos}^{2} \beta ) = 25 + {a}^{2} \\ \\ \\ 9 + 25 = 25 + {a}^{2} \\ \\ \\ {a}^{2} = 9 \\ \\ \\ a = 3 \: or \: - 3$

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