Let p = tan ( 5 9 cos ( 2 sin - 1 1 5 ) ) , q = sin - 1 ( sin 2 3 ) + cos - 1 ( cos ( 7 6 ) ) then the quadratic equation whose roots are p , sec q is (Here inverse trigonometric functions take principal values)

Question

Let $p = \tan (\frac{5 π}{9} \cos (2 \sin^{- 1} \frac{1}{\sqrt{5}})),$ $q = \sin^{- 1} (\sin \frac{2 π}{3}) + \cos^{- 1} (\cos (\frac{7 π}{6}))$ then the quadratic equation whose roots are $p$ , $\sec q$ is (Here inverse trigonometric functions take principal values)

Smart Achievers · Accepted Answer

(3)

$\cos (2 \sin^{- 1} \frac{1}{\sqrt{5}}) = \frac{3}{5},$ $\sin^{- 1} (\sin \frac{2 π}{3}) = \frac{π}{3},$ $\cos^{- 1} (\cos \frac{7 π}{6}) = 2 π - \frac{7 π}{6}$

Solution

Q.
Let $p = \tan (\frac{5 π}{9} \cos (2 \sin^{- 1} \frac{1}{\sqrt{5}})),$ $q = \sin^{- 1} (\sin \frac{2 π}{3}) + \cos^{- 1} (\cos (\frac{7 π}{6}))$ then the quadratic equation whose roots are $p$ , $\sec q$ is (Here inverse trigonometric functions take principal values)

1 $\sqrt{3} x^{2} - x + \sqrt{3} = 0$

2 $\sqrt{3} x^{2} - 4 x + 2 \sqrt{3} = 0$

3 $\sqrt{3} x^{2} - x - 2 \sqrt{3} = 0$

4 $x^{2} - \sqrt{3} x + = 0$

Ans.
(3)

$\cos (2 \sin^{- 1} \frac{1}{\sqrt{5}}) = \frac{3}{5},$ $\sin^{- 1} (\sin \frac{2 π}{3}) = \frac{π}{3},$ $\cos^{- 1} (\cos \frac{7 π}{6}) = 2 π - \frac{7 π}{6}$

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Solution

Q. Let p=tan(5π9cos(2sin-115)), q=sin-1(sin2π3)+cos-1(cos(7π6)) then the quadratic equation whose roots are p, sec q is (Here inverse trigonometric functions take principal values)

1 3x2-x+3=0

2 3x2-4x+23=0

3 3x2-x-23=0

4 x2-3x+=0

Ans. (3) cos(2sin-115)=35, sin-1(sin2π3)=π3, cos-1(cos7π6)=2π-7π6