Value of Isec 20 - circ 8- Whline12- circ sin 48-circsin 54 - circ - Whline- - - - - text triangle are in an arithmetic - textprogression and if a- b and c denote - -textthe lengths of sides opposite to A- B - - - - - and C respectively- then the value -text of the expression 1Which of the following options is only correct combination -

The correct option is A (I), (iv), (S)(I)cos ( sin^ 1 ( sin5 π/6 ) )=cos ( π/6 )=√(3)/2 (II)cos ( cos^ 1 ( sin7 π/6 ) )=cos ( π/3 )=1/2 (III)|cos ( tan^ 1 ( ...

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Byju's Answer

Standard XII

Mathematics

Sign of Trigonometric Ratios in Different Quadrants

value of Isec...

Question

$\begin{matrix} Column 1 & Column 2 & Column 3 f (x) = 3 x^{3} - 13 x^{2} + 14 x - 2 has (I) & cos ({sin}^{- 1} (sin \frac{5 π}{6})) & (i) & 1 & P & three roots α, β, γ, then value of | sin ({tan}^{- 1} α + {tan}^{- 1} β + {tan}^{- 1} γ) | = (II) & cos ({cos}^{- 1} (- sin \frac{7 π}{6})) = & (ii) & \frac{1}{\sqrt{2}} & Q & \frac{(\sqrt{3} csc 20^{\circ} - sec 20^{\circ})}{8} = (III) & ∣ ∣ cos {{tan}^{- 1} (- tan \frac{15 π}{4})} ∣ ∣ = & (iii) & \frac{1}{2} & R & 4 \sqrt{2} (sin 12^{\circ}) (sin 48^{\circ}) (sin 54^{\circ}) = (IV) & sin ({({cos}^{- 1} {\frac{1}{\sqrt{2}} (cos \frac{9 π}{10} - sin \frac{9 π}{10})} + \frac{7 π}{20})}^{- 1} \times π^{2}) & (iv) & \frac{\sqrt{3}}{2} & S & If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of sides opposite to A, B and C respectively, then the value of the expression \frac{1}{2} (\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A) is \end{matrix}$
Which of the following options is only correct combination ?

(I), (iv), (S)

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(I), (iv), (R)

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(I), (iii), (S)

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(I), (iii), (P)

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Solution

The correct option is A (I), (iv), (S)
$(I) cos ({sin}^{- 1} (sin \frac{5 π}{6})) = cos (\frac{π}{6}) = \frac{\sqrt{3}}{2} (II) cos ({cos}^{- 1} (- sin \frac{7 π}{6})) = cos (\frac{π}{3}) = \frac{1}{2} (III) ∣ ∣ cos ({tan}^{- 1} (- tan \frac{15 π}{4})) ∣ ∣ = ∣ ∣ cos (- \frac{π}{4}) ∣ ∣ = \frac{1}{\sqrt{2}} (IV) sin ({({cos}^{- 1} {\frac{1}{\sqrt{2}} (cos \frac{9 π}{10} - sin \frac{9 π}{10})} + \frac{7 π}{20})}^{- 1} \times π^{2}) = sin ({({cos}^{- 1} (cos \frac{23 π}{20}) + \frac{7 π}{20})}^{- 1} \times π^{2}) = sin ({(2 π - \frac{23 π}{20} + \frac{7 π}{20})}^{- 1} \times π^{2}) = sin ({(\frac{24 π}{20})}^{- 1} \times π^{2}) = sin (\frac{5 π}{6}) = \frac{1}{2}$
$(P) f (x) = 3 x^{3} - 13 x^{2} + 14 x - 2 α, β, γ are the roots of f (x) ∴ α + β + γ = \frac{13}{3} α β + β γ + γ α = \frac{14}{3} α β γ = \frac{2}{3}$

$∴ ∣ ∣ sin ({tan}^{- 1} α + {tan}^{- 1} β + {tan}^{- 1} γ) ∣ ∣ = ∣ ∣ ∣ ∣ sin ({tan}^{- 1} (\frac{\frac{13}{3} - \frac{2}{3}}{1 - \frac{14}{3}})) ∣ ∣ ∣ ∣ = ∣ ∣ sin ({tan}^{- 1} (- 1)) ∣ ∣ = \frac{1}{\sqrt{2}}$

$(Q) \frac{\sqrt{3} csc 20^{\circ} - sec 20^{\circ}}{8} = \frac{\frac{\sqrt{3}}{2} cos 20^{\circ} - \frac{1}{2} sin 20^{\circ}}{4 cos 20^{\circ} sin 20^{\circ}} = \frac{sin 60^{\circ} cos 20^{\circ} - cos 60^{\circ} sin 20^{\circ}}{2 (2 sin 20^{\circ} cos 20^{\circ})} = \frac{sin 40^{\circ}}{2 sin 40^{\circ}} = \frac{1}{2}$

$(R) 4 \sqrt{2} (sin 12^{\circ}) (sin 48^{\circ}) (sin 54^{\circ}) = 2 \sqrt{2} (2 sin 12^{\circ} sin 48^{\circ}) sin 54^{\circ} = 2 \sqrt{2} [cos 36^{\circ} - cos 60^{\circ}] sin 54^{\circ} = 2 \sqrt{2} [cos 36^{\circ} - \frac{1}{2}] sin 54^{\circ} = 2 \sqrt{2} [{(\frac{\sqrt{5} + 1}{4})}^{2} - \frac{1}{2} (\frac{\sqrt{5} + 1}{4})] = \frac{1}{\sqrt{2}}$

$(S) Since a, b, c are in A.P. ∴ ∠ B = 60^{\circ}$
$∴ \frac{1}{2} (\frac{a}{c} sin 2 C + \frac{c}{a} (sin 2 A) = \frac{1}{2} (\frac{a}{c} (2 sin C cos C) + \frac{c}{a} (2 sin A cos A)) = \frac{sin C}{c} (a cos C) + \frac{sin A}{a} (c cos A) = \frac{1}{2 R} (a cos C + c cos A) [∵ \frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R] = \frac{b}{2 R} = sin B = \frac{\sqrt{3}}{2}$

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Similar questions

\begin{matrix} Column 1 & Column 2 & Column 3 f (x) = 3 x^{3} - 13 x^{2} + 14 x - 2 has (I) & cos ({sin}^{- 1} (sin \frac{5 π}{6})) & (i) & 1 & P & three roots α, β, γ, then value of | sin ({tan}^{- 1} α + {tan}^{- 1} β + {tan}^{- 1} γ) | = (II) & cos ({cos}^{- 1} (- sin \frac{7 π}{6})) = & (ii) & \frac{1}{\sqrt{2}} & Q & \frac{(\sqrt{3} csc 20^{\circ} - sec 20^{\circ})}{8} = (III) & ∣ ∣ cos {{tan}^{- 1} (- tan \frac{15 π}{4})} ∣ ∣ = & (iii) & \frac{1}{2} & R & 4 \sqrt{2} (sin 12^{\circ}) (sin 48^{\circ}) (sin 54^{\circ}) = (IV) & sin ({({cos}^{- 1} {\frac{1}{\sqrt{2}} (cos \frac{9 π}{10} - sin \frac{9 π}{10})} + \frac{7 π}{20})}^{- 1} \times π^{2}) & (iv) & \frac{\sqrt{3}}{2} & S & If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of sides opposite to A, B and C respectively, then the value of the expression \frac{1}{2} (\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A) is \end{matrix}

Which of the following options is only INCORRECT combination ?

\begin{matrix} Column 1 & Column 2 & Column 3 f (x) = 3 x^{3} - 13 x^{2} + 14 x - 2 has (I) & cos ({sin}^{- 1} (sin \frac{5 π}{6})) & (i) & 1 & P & three roots α, β, γ, then value of | sin ({tan}^{- 1} α + {tan}^{- 1} β + {tan}^{- 1} γ) | = (II) & cos ({cos}^{- 1} (- sin \frac{7 π}{6})) = & (ii) & \frac{1}{\sqrt{2}} & Q & \frac{(\sqrt{3} csc 20^{\circ} - sec 20^{\circ})}{8} = (III) & ∣ ∣ cos {{tan}^{- 1} (- tan \frac{15 π}{4})} ∣ ∣ = & (iii) & \frac{1}{2} & R & 4 \sqrt{2} (sin 12^{\circ}) (sin 48^{\circ}) (sin 54^{\circ}) = (IV) & sin ({({cos}^{- 1} {\frac{1}{\sqrt{2}} (cos \frac{9 π}{10} - sin \frac{9 π}{10})} + \frac{7 π}{20})}^{- 1} \times π^{2}) & (iv) & \frac{\sqrt{3}}{2} & S & If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of sides opposite to A, B and C respectively, then the value of the expression \frac{1}{2} (\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A) is \end{matrix}

Which of the following options is only correct combination ?

Q. If the angles

A, B

and

C

of a triangle are in an arithmetic progression and if

a, b

and

c

denote the lengths of the sides opposite to

A, B

and

C

respectively, then the value of the expression

\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A

Q. If the angles

A, B

and

C

of a triangle are in arithmetic progression and if

a, b

and

c

denote the lengths of the sides opposite to

A, B

and

C

respectively, then the value of the expression

\frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A

is:

$\begin{matrix} C o l u m n I & C o l u m n 2 & C o l u m n 3 (I) & c o s (s i n^{- 1} (s i n \frac{5 π}{6})) = & (i) & 1 & (P) & f (x) = 3 x^{3} - 13 x^{2} + 14 x - 2 h a s t h r e e r o o t s α, β, γ, t h e n v a l u e o f | s i n (t a n^{- 1} α + t a n^{- 1} β + t a n^{- 1} γ) | = (I I) & c o s (c o s^{- 1} (- s i n \frac{7 π}{6})) = & (i i) & \frac{1}{\sqrt{2}} & (Q) & \frac{(\sqrt{3} c o s e c 20^{\circ} - s e c 20^{\circ})}{8} = (I I I) & ∣ ∣ c o s {t a n^{- 1} (t a n \frac{15 π}{4})} ∣ ∣ = & (i i i) & \frac{1}{2} & (R) & 4 \sqrt{2} (s i n 12^{\circ}) (s i n 48^{\circ}) (s i n 54^{\circ}) = (I V) & s i n ({(c o s^{- 1} {\frac{1}{\sqrt{2}} (c o s \frac{9 π}{10} - s i n \frac{9 π}{10})} + \frac{7 π}{20})}^{- 1} \times π^{2}) & (i v) & \frac{\sqrt{3}}{2} & (S) & I f t h e a n g l e s A, B a n d C o f a t r i a n g l e a r e i n a n a r i t h m e t i c p r o g r e s s i o n a n d i f a, b a n d c d e n o t e t h e l e n g t h s o f s i d e s o p p o s i t e t o A, B a n d C r e s p e c t i v e l y, t h e n t h e v a l u e o f t h e e x p r e s s i o n \frac{1}{2} (\frac{a}{c} s i n 2 C + \frac{c}{a} s i n 2 A) i s \end{matrix}$

Which of the following options is only INCORRECT combination?

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