Question

Solve for x and y 12sinx+5cosx = 2y^2-8y+21

Answer 1

you can rewrite left hand side as
\[\sf 12\sin x + 5 \cos x = \langle 5, 12\rangle \cdot \langle \cos x, \sin x\rangle = \sqrt{5^2+12^2}\cos\left(x-\arctan\left( \frac{12}{5}\right)\right)\]

Answer 2

try completing the square for right hand side

Answer 3

\[\sf 2y^2 - 8y+21 = 2(y-2)^2 + 13\]

Answer 4

\[\sf 13\cos\left(x-\arctan\left( \frac{12}{5}\right)\right) = 2(y-2)^2+13 \]

Notice that the maximum value of left hand side is 13 and the minimum value of right hand side is 13. What can you conclude from that ?

Answer 5

thank you

Answer 6

yw! so whats your solution for x and y ?