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Integration of Trigonometric Functions

Solve for x...

Question

Solve for $x$ and $y$ :
$12 s i n x + 5 c o s x = 2 y^{2} - 8 y + 21$

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Solution

Given, $12 sin x + 5 cos x = 2 y^{2} - 8 y + 21$
As $- 13 \leq 12 sin x + 5 cos x \leq 13$
${2 y}^{2} - 8 y + 21 = - 13$ or ${2 y}^{2} - 8 y + 21 = 13$
${2 y}^{2} - 8 y + 34 = 0$ or ${2 y}^{2} - 8 y + 8 = 0$
$y^{2} - 4 y + 17 = 0$ or $y^{2} - 4 y + 4 = 0$
imaginary roots or ${(y - 2)}^{2} = 0$
$y = 2$

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12 sin x + 5 cos x = 2 y^{2} - 8 y + 21

to get the values of

n

y

x

y

satisfies the equation

12 sin x + 5 cos x = 2 y^{2} - 8 y + 21

, then the value of

12 cot (\frac{x y}{2})

x

y

are the solutions of the equation

12 sin x + 5 cos x = 2 y^{2} - 8 y + 21

3864 cot (\frac{x y}{2})

a = \frac{1}{5 cos x + 12 sin x}

, then for all real

x

y = {tan}^{- 1} [\frac{5 cos x - 12 sin x}{12 cos x + 5 sin x}]

\frac{d y}{d x} =

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