MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Trigonometric Ratios for Sum of Two Angles

If sinα + s...

Question

If $sin α + sin β = a, cos α + cos β = b$ , then $sin (α + β)$ equals

A

a b

No worries! We‘ve got your back. Try BYJU‘S free classes today!

B

a + b

No worries! We‘ve got your back. Try BYJU‘S free classes today!

C

\frac{2 a b}{a^{2} - b^{2}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D

\frac{2 a b}{a^{2} + b^{2}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $\frac{2 a b}{a^{2} + b^{2}}$
$sin α + sin β = a, cos α + cos β = b$

Squaring and adding
$a^{2} + b^{2} = {sin}^{2} α + {sin}^{2} β + 2 sin α sin β + {cos}^{2} α + {cos}^{2} β + 2 cos α cos β$
$a^{2} + b^{2} = 2 + 2 cos (α - β) . . . (1)$

$2$ times multiplying both the expression we get
$\Rightarrow 2 a b = 2 sin α cos α + 2 sin α cos β + 2 sin β cos α + 2 sin β cos β$
$2 a b = sin 2 α + sin 2 β + 2 sin (α + β) . . . (2)$

dividing $(2)$ by $(1)$

$\Rightarrow \frac{sin 2 α + sin 2 β + 2 sin (α + β)}{2 + 2 cos (α - β)} = \frac{2 a b}{a^{2} + b^{2}}$

Using the formula $sin c + sin d = 2 sin (\frac{c + d}{2}) \times cos (\frac{c - d}{2})$ , we get

$\Rightarrow \frac{2 sin (α + β) cos (α - β) + 2 sin (α + β)}{2 + 2 cos (α - β)} = \frac{2 a b}{a^{2} + b^{2}}$

$\Rightarrow \frac{sin (α + β) (1 + cos (α - β))}{(1 + cos (α - β))} = \frac{2 a b}{a^{2} + b^{2}}$

$\Rightarrow sin (α + β) = \frac{2 a b}{a^{2} + b^{2}}$

Suggest Corrections

thumbs-up

0

Similar questions

s i n α + s i n β = a

c o s α + c o s β = b,

s i n (α + β) = \frac{2 a b}{a^{2} + b^{2}}

c o s (α + β) = \frac{b^{2} - a^{2}}{b^{2} + a^{2}}

If $s i n α + s i n β = a a n d c o s α + c o s β = b$ , prove that

(i) $s i n (α + β) = \frac{2 a b}{a^{2} + b^{2}}$

(ii) $c o s (α - β) = \frac{a^{2} + b^{2} - 2}{2}$

If sin α + sin β = a

cos α + cos β = b

, then the value of

sin (α + β)

sin α + sin β = a

cos α + cos β = b

, then the value of

sin α + β = \frac{m a b}{a^{2} + b^{2}}

m

sin α + sin β = a

cos α + cos β = b

tan \frac{α - β}{2} = \pm \sqrt{\frac{x - a^{2} - b^{2}}{a^{2} + b^{2}}}

x

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Trigonometric Ratios for Sum of Two Angles

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon