Question

The equation ${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30$ has two roots in the

• A. I quadrant
• B. II quadrant
• C. III quadrant
• D. IV quadrant

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A I quadrant
B II quadrant
C III quadrant
D IV quadrant

Let ${81}^{{\sin }^{2}x}=y.$

Then ${81}^{{\cos }^{2}x}={81}^{1-{\sin }^{2}x}=81.y^{-1}$,

so that the given equation is equivalent to
$y+81y^{-1}=30$

$y^2-30y+81=0$

solve the equation in $y$

$\Rightarrow y=3$ or $y=27$

$\Rightarrow {81}^{{\sin }^{2}x}=3$ or ${81}^{{\sin }^{2}x}=27$

$\Rightarrow 3^{4{\sin }^{2}x}=3$ or $3^{4{\sin }^{2}x}=3^3$

$\Rightarrow {\sin }^{2}x=\frac{1}{4}$ or ${\sin }^{2}x=\frac{3}{4}$

$\sin x=\pm \frac{1}{2}$ or $\sin x=\pm \frac{\sqrt{3}}{2}$

which gives two solutions of the given equation in each quadrant.