Question

if
$m \cos( \alpha ) - n \sin( \alpha ) = p \:$
then prove that
$m \sin( \alpha ) + n \cos( \alpha ) = + - \sqrt{m {}^{2} + n {}^{2} - p {}^{2} }$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Identities Used :-

$\boxed{ \sf{ \: {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy}}$

$\boxed{ \sf{ \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}}$

$\boxed{ \sf{ \: {sin}^{2}x + {cos}^{2}x = 1}}$

$\boxed{ \sf{ \: {sin}^{2}x = 1 - {cos}^{2}x}}$

$\boxed{ \sf{ \: {cos}^{2}x = 1 - {sin}^{2}x}}$

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\:mcos \alpha - nsin\alpha = p$

On squaring both sides, we get

$\rm :\longmapsto\:(mcos \alpha - nsin\alpha)^{2} = p^{2}$

$\rm :\longmapsto\: {m}^{2} {cos}^{2}\alpha + {n}^{2} {sin}^{2}\alpha - 2mncos\alpha sin\alpha = {p}^{2}$

$\rm :\longmapsto\: {m}^{2}(1 - {sin}^{2}\alpha) + {n}^{2}(1 - {cos}^{2}\alpha) - 2mncos\alpha sin\alpha = {p}^{2}$

$\rm :\longmapsto\: {m}^{2} - {m}^{2} {sin}^{2}\alpha+ {n}^{2}- {n}^{2} {cos}^{2}\alpha- 2mncos\alpha sin\alpha = {p}^{2}$

$\rm :\longmapsto\: - {m}^{2} {sin}^{2}\alpha- {n}^{2} {cos}^{2}\alpha- 2mncos\alpha sin\alpha = {p}^{2} - {m}^{2} - {n}^{2}$

$\rm :\longmapsto\: {m}^{2} {sin}^{2}\alpha + {n}^{2} {cos}^{2}\alpha + 2mncos\alpha sin\alpha = - {p}^{2} + {m}^{2} + {n}^{2}$

$\rm :\longmapsto\:(msin\alpha + ncos\alpha )^{2} = {m}^{2} + {n}^{2} - {p}^{2}$

$\rm :\longmapsto\:msin\alpha + ncos\alpha = \: \pm \: \sqrt{{m}^{2} + {n}^{2} - {p}^{2}}$

Hence, proved

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1