Question

If acosx+bsinx=4 and asinx-bcosx=3, then a2+b2is:


7

12

25

can t find

Answer 1

Your answer is 25. First of all squaring both equations and simply adding them.

Answer 2

Given, a cosx + b sinx = 4       ...( i )

           a sinx - b cosx = 3       ...( ii )

Square on both sides on ( i )

⇒ ( a cosx + b sinx )^2 = 4^2

From the identities of factorization, we know ( a + b )^2 = a^2 + b^2 + 2ab

⇒ ( a cosx )^2 + ( b sinx )^2 + 2( a cox . b sinx ) = 16

⇒ a^2 cos^2 x + b^2 sin^2 x + 2ab.cosx.sinx = 16     ...( iii )

Square on both sides on ( ii )

⇒ ( a sinx - b cosx )^2 = 3^2

From the identities of factorization, we know ( a - b )^2 = a^2 + b^2 - 2ab

⇒ ( a sinx )^2 + ( b cosx )^2 - 2( b cox . a sinx ) = 16

⇒ a^2 sin^2 x + b^2 cos^2 x - 2ab.cosx.sinx = 16     ...( iv )

Now adding ( iii ) and ( iv ) to each other : -

⇒ a^2 cos^2 x + b^2 sin^2 x + 2ab.cosx.sinx + a^2 sin^2 x + b^2 cos^2 x - 2ab.cosx.sinx = 16 + 9

⇒ a^2 cos^2 x + a^2 sin^2 x + b^2 sin^2 x + b^2 cos^2 x = 25

⇒ a^2( cos^2 x + sin^2 x ) + b^2( sin^2 x + cos^2 x ) = 25

From the identities of trigonometry we know, sin^2 A + cos^2 A = 1

∴ a^2( 1 ) + b^2( 1 ) = 25

⇒ a^2 + b^2 = 25

Hence the value of a^2 + b^2 is 25.

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