Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

prove that:sinAcosB+cosAsinB=sin(A+B)sin(A-B)

prove that:sinAcosB+cosAsinB=sin(A+B)sin(A-B)

Grade:11

1 Answers

SAGAR SINGH - IIT DELHI
879 Points
10 years ago

Dear kiran,

Given the functions (sinα, cosα, sinβ and cos β), we seek a formula that expresses sin(α+β).

ABC which has an angle α
ACD which  "   "    "  β
The long side ("hypotenuse') of ACD is AD=R. Therefore

DC = R sin β
AC = R cos β
Similarly

BC = AC sin α = R cos β sin α
AB = AC cos α = R cos β cos α

The triangle ADF is right-angled and has the angle (α+β). Therefore

R sin (α+β) = DF
R cos (α+β) = AF

Start by deriving the sine:
R sin (α+β) = DF  =  EF + DE  =  BC + DE

 Note in the drawing the two head-to-head angles marked with double lines: like all such angles, they must be equal. Each of them is one of the two sharp ("acute") angles in its own right-angled triangle. Since the sharp angles in such a triangle add up to 90 degrees, the other two sharp angles must be equal. This justifies marking the angle near D as α, as drawn in the figure.
In the right-angled triangle CED

DE = DC cos α = R sin β cos α
EC = DC sin α = R sin β sin α
Earlier it was already shown that
BC = R cos β sin α
AB = R cos β cos α
Therefore
R sin (α+β)  =  BC+DE  =  R cos β sin α + R sin β cos α

Cancelling R we have

  sin (α+β)  =  sin α cos β  + cos α sin β

 

We are all IITians and here to help you in your IIT JEE preparation.

All the best.

 If you like this answer please approve it....

win exciting gifts by answering the questions on Discussion Forum

 

Sagar Singh

B.Tech IIT Delhi

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free