Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
cos(x)+cos(y)+cos(z)=0 and sin(x)+sin(y)+sin(z)=0. prove that cos(x-y)+cos(y-z)+cos(z-x)=-3/2
Hi Aritra,
Let's see how to solve this easily using Complex Numbers:
Let z1=cosx+isinx, z2=cosy+isiny, z3=cosz+isinz.
As Re(z1+z2+z3)=0 and Img(z1+z2+z3)=0,
we have z1+z2+z3=0.
Also as |z|=1 for all complex numbers, 1/z = z' (where ' denotes the conjugate, which is z bar).
so 1/z1 = z1'........ 1/z2 = z2'......... 1/z3 = z3'.
Next cos(x-y)+cos(y-z)+(z-x) = Re(z1/z2 + z2/z3 + z3/z1) = Re (z1z2' + z2z3' + z3z1')------(1).
Re(z) = 1/2(z+z')
So (1) = (1/2)*(z1z2'+z2z3'+z3z1'+z1'z2+z2'z3+z3'z1)
= (1/2)*[z2'(z1+z3) + z3'(z1+z2) + z1'(z2+z3)]
= (1/2)*[z2'(-z2) + z3'(-z3) + z1'(-z1)]
= (1/2)*[-|z2|2-|z3|2-|z1|2]
= -3/2.
And hence proved. Hope that helps.
All the best.
Regards,
Ashwin (IIT Madras).
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Win Gift vouchers upto Rs 500/-
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !