First slide

Trigonometric equations

Question

Values of x and y satisfying the equation $\sin^{7} y = | x^{3} - x^{2} - 9 x + 9 | + | x^{3} - x^{2} - 4 x + 4 | + \sec^{2} 2 y + \cos^{4} y$ are

Moderate

A

$x = 1, y = n π, n \in I$
B

$x = 1, y = 2 n π + \frac{π}{2}, n \in I$
C

$x = 1, y = 2 n π, n \in I$
D

none of the above

Solution

$\sin^{7} y = | x^{3} - x^{2} - 9 x + 9 | + | x^{3} - x^{2} - 4 x + 4 |$ $+ \sec^{2} 2 y + \cos^{4} y$
For $x = 1$

$\sin^{7} y = \sec^{2} 2 y + \cos^{4} y$

$\Rightarrow \sin^{7} y \cos^{2} 2 y = 1 + \cos^{4} y \cos^{2} 2 y$

Since, $L H S \leq 1.$ and $R H S \geq 1$

Which is possible only when

$∴ \sin^{7} y \cos^{2} 2 y = 1$

$\Rightarrow \sin^{7} y = 1 a n d \cos^{2} 2 y = 1; y = \frac{π}{2}$

General value of y is $2 n π + \frac{π}{2}$

Hence, $x = 1$ and $y = 2 n π + \frac{π}{2}, n \in I$

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