Question

If the angles of a triangle $ABC$ satisfy the equation ${81}^{si{n}^{2}x}+{81}^{co{s}^{2}x}=30$, then the triangle can be

• A. equilateral
• B. isosceles
• C. obtuse angled
• D. right angled

Answer 1

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

The correct options are
A isosceles
B equilateral
C obtuse angled

Given, ${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30$

${81}^{{\sin }^{2}x}+{81}^{1-{\sin }^{2}x}=30$

$\Rightarrow y+\frac{81}{y}=30$, where $({81}^{{\sin }^{2}x}=y$)

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

$\Rightarrow y={81}^{{\sin }^{2}x}=3^{4{\sin }^{2}x}=3or27$

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

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

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

Hence triangle can be equilateral, obtuse or isosceles but not right angle.