Let - - be such that -3 - If sin-sin-21-65 and cos-cos-27-65- then the value of cos-2 is

The correct option is A Given, sinα+sinβ= 21/65and cosα+cosβ= 27/65 On squaring and adding the above equations, we get sin^2 α+sin^2 β+2sin α sin β + cos^2α ...

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Standard XI

Mathematics

Trigonometric Equations

Let α, β be s...

Question

$L e t α, β b e s u c h t h a t π < α - β < 3 π . I f s i n α + s i n β = \frac{- 21}{65} a n d c o s α + c o s β = \frac{- 27}{65}, t h e n t h e v a l u e o f c o s \frac{α - β}{2} i s$

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6/65

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-6/65

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Solution

The correct option is A

$G i v e n, s i n α + s i n β = \frac{- 21}{65} a n d c o s α + c o s β = \frac{- 27}{65} O n s q u a r i n g a n d a d d i n g t h e a b o v e e q u a t i o n s, w e g e t s i n^{2} α + s i n^{2} β + 2 s i n α s i n β + c o s^{2} α + c o s^{2} β + 2 c o s α c o s β = {(\frac{21}{65})}^{2} + {(\frac{27}{65})}^{2} \Rightarrow 2 + 2 (s i n α s i n β + c o s α c o s β) = \frac{1170}{4225} \Rightarrow 2 + 2 c o s (α - β) = \frac{1170}{4225} \Rightarrow 4 c o s^{2} (\frac{α - β}{2}) \Rightarrow c o s^{2} (\frac{α - β}{2}) = \frac{9}{130} \Rightarrow c o s (\frac{α - β}{2}) = - \frac{3}{\sqrt{130}} ∵ π < α - β < 3 π$

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L e t α, β b e s u c h t h a t π < α - β < 3 π . I f s i n α + s i n β = \frac{21}{65} a n d c o s α + c o s β = - \frac{27}{65}, t h e n t h e v a l u e o f c o s \frac{α - β}{2} i s

Q. Let

α

and

β

be such that

π < α - β < 2 π

. If

sin α + sin β = - \frac{21}{65}

and

cos α + cos β = \frac{17}{65}

, then the value of

cos \frac{α - β}{2}

Q. Let

z_{1}

and

z_{2}

be two complex numbers with

α

and

β

as their principal arguments such that

α + β > π,

then principal arg

(z_{1} z_{2})

is given by

Let $a_{n}$ be the $n^{t h}$ term of the G.P. of positive numbers. Let $\sum_{n = 1}^{100} a_{2 n} = α$ and $\sum_{n = 1}^{100} a_{2 n - 1} = β$ , such that such that $α \neq β$ . Prove that the common ratio of the G.P. is $\frac{α}{β}$ .

L e t α, β b e s u c h t h a t π < α - β < 3 π . I f s i n α + s i n β = \frac{21}{65} a n d c o s α + c o s β = - \frac{27}{65}, t h e n t h e v a l u e o f c o s \frac{α - β}{2} i s

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Let α,β be such that π < α−β < 3π. If sin α+sin β=−2165 and cos α+cosβ=−2765, then the value of cos α−β2is

$L e t α, β b e s u c h t h a t π < α - β < 3 π . I f s i n α + s i n β = \frac{- 21}{65} a n d c o s α + c o s β = \frac{- 27}{65}, t h e n t h e v a l u e o f c o s \frac{α - β}{2} i s$