Trigno Problem?

Question

if the angles A,B,C of a triangle are in A.P.such dat sin(2A+B)=1/2
then sin(B+2C)=?

Answer 1

math_kp's answer is correct, but if you do want A, B and C in arithmetic progression then let B = A + n and C = A + 2n.
Then, A + B + C = A + (A + n) + (A + 2n) = 3A + 3n = 180º.
Thus, A + n = 60º.
Now sin(2A + B) = sin(2A + A + n) = sin(3A + n) = 1/2.
The principal value of 3A + n is 30º, but this won't work if A + n = 60º. So try the next value for 3A + n which is 150º, because sin(X) = sin(180º - X), in general.
Now we have the 2 equations, A + n = 60 and 3A + n = 150, from which we can extract A = 45º and n = 15º.
Thus we get the values in A.P. as A = 45º, B = 60º and C = 75º.
Then, sin(B+2C) = sin(60º + 2*75º) = sin(210º) = -0.5.

Answer 2

i'm not familiar with the term A.P. with trig. Only A.P. (arithmetic progression)

However, sin (30 degrees) = 1/2
30 = 2a + b

sin (2a + b) + sin (b + 2c) = sin (2a +2b +2c)

Wouldn't sin (2a +2b +2c) = sin 360 degrees

let us know if that is right or not. good luck.

Only just thought of that and it's been two years since i've done any of these problems.

So sin (b+2c) = sin 330 degrees.

which is -0.5

Answer 3

A B C are in AP

But they need not be because even if they are not result is same

A+B+C = 180
so 2A + 2B +2C = 360

so B+2C = (360-(B+2A)

so sin (B+2C) = - sin (B+ 2A) = -.5
we have no where used they are in AP

Answer 4

Sin 30 = 1/2

Therefore 2A+B = 30 degrees

By the way what is A.P

Answer 5

A+B+C=180....1
sin[2A+B]=1/2
2A+B=30Deg...2
2A+2B+2C=360 (from 1)...3
from 2&3 we have
B+2C=330
sin[B+2C]=sin[330]
we know sinz=sin[360-z]
sin330=sin[360-330]=sin30=1/2

Answer 6

so (2A+B)=30*
A+B+C=180*
2A+2B+2C=360*
B+2C=330*
sin (B+2C)=-1/2