Question

if 10 cosA - 11 Sin A = 12, Show that 10 SinA + 11 CosA = ± √78​

Answer 1

We have:

10 $\cos A$ - 11 $\sin A$ = 12                   ........ (1)

Let 10$\sin A$ + 11 $\cos A$ = x              ........ (2)

We have to show that, 10$\sin A$ + 11 $\cos A$ = $\sqrt{77}$.

Solution:

Squaring and adding equations (1) and (2), we get

$(10\cos A -11\sin A)^2+(10\sin A +11\cos A)^2$= $12^{2} +x^2$

⇒ $100\cos^2 A +121\sin^2 A-2\sin A\cos A+100\sin^2 A +121\cos^2 A+2\sin A\cos A$= $144 +x^2$

⇒ $100(\sin^2 A+\cos^2 A) +121(\sin^2 A+\cos^2 A)$= $144 +x^2$

⇒ $100(1) +121(1)$= $144 +x^2$

⇒ 100 + 121 = $144 +x^2$

⇒ 221 = $144 +x^2$

⇒ $x^{2}$ = 221 -144

⇒ $x^{2}$ = 77

⇒ x = ± $\sqrt{77}$

Thus, 10$\sin A$ + 11 $\cos A$ = $\sqrt{77}$, shown.

Answer 2

SOLUTION

GIVEN

$\sf{10 \cos A - 11 \sin A = 12}$

TO PROVE

$\sf{10 \sin A + 11 \cos A = \pm \: \sqrt{77} }$

PROOF

Here it is given that

$\sf{10 \cos A - 11 \sin A = 12}$

Squaring both sides we get

$\sf{{(10 \cos A - 11 \sin A)}^{2} = 12}$

$\sf{ \implies100 \: {\cos}^{2} A - 220 \: \cos A \: \sin A + 121 \: { \sin}^{2} A = 144}$

$\sf{ \implies100 \:(1 - {\sin}^{2} A) - 220 \: \cos A \: \sin A + 121 \: (1 - { \cos}^{2} A )= 144}$

$\sf{ \implies100 - 100 \: {\sin}^{2} A - 220 \: \cos A \: \sin A + 121 - 121 \: { \cos}^{2} A )= 144}$

$\sf{ \implies100 \: {\sin}^{2} A + 220 \: \cos A \: \sin A + 121 \: { \cos}^{2} A = 100 + 121 - 144}$

$\sf{ \implies100 \: {\sin}^{2} A + 220 \: \cos A \: \sin A + 121 \: { \cos}^{2} A = 77}$

$\implies \sf{{(10 \sin A + 11 \cos A) }^{2} = 77 }$

$\implies \sf{(10 \sin A + 11 \cos A) = \pm \: \sqrt{77} }$

Hence proved

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