Let- -be such that- - - - - - 3 - If sin -sin -2165 and cos - - cos - -2765 then the value ofcos -2 is

Given that sin α+sin β = 2165, cos α + cos β = 2765 Now (sin α+sin β)^2+(cos α + cos β)^2= ( 2165 )^2+ ( 2765 )^2 ⇒ 2+2 sin α sin β+2 cos α cos β=44165^2 ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Compound Angles: Cosine Function

Letα, βbe suc...

Question

Let $α, β$ be such that $π < α - β < 3 π$ . If $s i n α + s i n β = \frac{21}{65} a n d c o s α + c o s β = - \frac{27}{65}$
then the value of $c o s \frac{α - β}{2} i s$

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Solution

Given that
$s i n α + s i n β = - \frac{21}{65}, c o s α + c o s β = - \frac{27}{65} N o w (s i n α + s i n β)^{2} + (c o s α + c o s β)^{2} = {(\frac{- 21}{65})}^{2} + {(\frac{- 27}{65})}^{2} \Rightarrow 2 + 2 s i n α s i n β + 2 c o s α c o s β = \frac{441}{65^{2}} + \frac{729}{65^{2}} \Rightarrow 2 + 2 [c o s (α - β)] = \frac{1170}{(65)^{2}} \Rightarrow 2.2 c o s^{2} (\frac{α - β}{2}) = \frac{1170}{(65)^{2}} \Rightarrow c o s (\frac{α - β}{2}) = \frac{3 \sqrt{130}}{130} \Rightarrow c o s (\frac{α - β}{2}) = \pm \frac{3}{\sqrt{130}} {g i v e n t h a t π < α - β < 3 π, ∴ \frac{π}{2} < \frac{α - β}{2} < \frac{3 π}{2}} T h e r e f o r e c o s (\frac{α - β}{2}) = \frac{- 3}{\sqrt{130}}$

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Similar questions

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

If
$s i n α + s i n β + s i n γ = 0 = c o s α + c o s β + c o s γ$
then value of
$c o s (α - β) + c o s (β - γ) + c o s (γ - α) i s$

Q. Let

α, β

be such that

π < α - β < 3 π

. If

sin α + sin β = - \frac{21}{65} a n d cos α + cos β = - \frac{27}{65}

, then value of

cos \frac{α - β}{2}

equals

Q. Let α, β be such that π < α – β < 3π. If sin α + sin β =

- \frac{21}{65}

and cos α + cos β =

- \frac{27}{65}

, then the

\cos \frac{α - β}{2}

=
(a)

- \frac{3}{\sqrt{130}}

(b)

\frac{3}{\sqrt{130}}

(c)

\frac{6}{65}

(d)

- \frac{6}{35}

Q. If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).

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Let α, βbe such that π<α−β<3π. If sin α+sin β=2165 and cos α+cos β=−2765 then the value of cosα−β2 is

Let $α, β$ be such that $π < α - β < 3 π$ . If $s i n α + s i n β = \frac{21}{65} a n d c o s α + c o s β = - \frac{27}{65}$
then the value of $c o s \frac{α - β}{2} i s$