+0  
 
0
244
2
avatar

prove sin(3/2 x)*sin x / sin(1/2 x) = sin x + sin 2x

Guest Jul 28, 2017
 #1
avatar
0

Verify the following identity:
sin((3 x)/2) (sin(x))/(sin(x/2)) = sin(x) + sin(2 x)

Multiply both sides by sin(x/2):
sin(x) sin((3 x)/2) = ^?sin(x/2) (sin(x) + sin(2 x))

sin(x) sin((3 x)/2) = 1/2 (cos(x - (3 x)/2) - cos(x + (3 x)/2)) = 1/2 (cos(-x/2) - cos((5 x)/2)):
(cos(-x/2) - cos((5 x)/2))/(2) = ^?sin(x/2) (sin(x) + sin(2 x))

Use the identity cos(-x/2) = cos(x/2):
(cos(x/2) - cos((5 x)/2))/(2) = ^?sin(x/2) (sin(x) + sin(2 x))

sin(x/2) (sin(x) + sin(2 x)) = sin(x/2) sin(x) + sin(x/2) sin(2 x):
(cos(x/2) - cos((5 x)/2))/(2) = ^?sin(x/2) sin(x) + sin(x/2) sin(2 x)

sin(x/2) sin(x) = 1/2 (cos(x/2 - x) - cos(x/2 + x)) = 1/2 (cos(-x/2) - cos((3 x)/2)):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(-x/2) - cos((3 x)/2))/(2) + sin(x/2) sin(2 x)

Use the identity cos(-x/2) = cos(x/2):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(x/2) - cos((3 x)/2))/(2) + sin(x/2) sin(2 x)

(cos(x/2) - cos((3 x)/2))/(2) = 1/2 cos(x/2) - 1/2 cos((3 x)/2):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(x/2))/(2) - (cos((3 x)/2))/(2) + sin(x/2) sin(2 x)

sin(x/2) sin(2 x) = 1/2 (cos(x/2 - 2 x) - cos(x/2 + 2 x)) = 1/2 (cos(-(3 x)/2) - cos((5 x)/2)):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(x/2))/(2) - (cos((3 x)/2))/(2) + (cos(-(3 x)/2) - cos((5 x)/2))/(2)

Use the identity cos(-(3 x)/2) = cos((3 x)/2):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(x/2))/(2) - (cos((3 x)/2))/(2) + (cos((3 x)/2) - cos((5 x)/2))/(2)

(cos((3 x)/2) - cos((5 x)/2))/(2) = 1/2 cos((3 x)/2) - 1/2 cos((5 x)/2):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(x/2))/(2) - (cos((3 x)/2))/(2) + (cos((3 x)/2))/(2) - (cos((5 x)/2))/(2)

(cos(x/2))/(2) - (cos((3 x)/2))/(2) + (cos((3 x)/2))/(2) - (cos((5 x)/2))/(2) = 1/2 cos(x/2) - 1/2 cos((5 x)/2):
(cos(x/2) - cos((5 x)/2))/(2) = ^?(cos(x/2))/(2) - (cos((5 x)/2))/(2)

1/2 (cos(x/2) - cos((5 x)/2)) = 1/2 cos(x/2) - 1/2 cos((5 x)/2):
1/2 cos(x/2) - 1/2 cos((5 x)/2) = ^?1/2 cos(x/2) - 1/2 cos((5 x)/2)

The left hand side and right hand side are identical:
Answer: | (identity has been verified)

Guest Jul 28, 2017
 #2
avatar
+1

Using the trig identity

\(\displaystyle \sin A+\sin B=2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)\) ,

 

RHS = \(\displaystyle \sin 2x + \sin x = 2 \sin \left( \frac{3x}{2}\right)\cos\left(\frac{x}{2}\right)= \frac{2 \sin \left( \frac{3x}{2}\right)\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} =\frac{\sin \left( \frac{3x}{2}\right)\sin\left(x\right)}{\sin\left(\frac{x}{2}\right)} \) .

 

Tiggsy

Guest Jul 28, 2017

13 Online Users

avatar
avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.