If sinA + sinB = a and cosA + cosB = b, find the value of tanA-B/2
Galat answer h chutiye
To find the value of tan(A - B/2), we can use the trigonometric identity:
tan(A - B/2) = (sin(A - B/2)) / (cos(A - B/2))
So, let's start by finding the values of sin(A - B/2) and cos(A - B/2).
To do that, we can use the following trigonometric identities:
sin(A - B/2) = sinA * cos(B/2) - cosA * sin(B/2)
cos(A - B/2) = cosA * cos(B/2) + sinA * sin(B/2)
Now, let's substitute the given values sinA + sinB = a and cosA + cosB = b into the above expressions.
sin(A - B/2) = sinA * cos(B/2) - cosA * sin(B/2)
= (sinA + sinB) * cos(B/2) - (cosA + cosB) * sin(B/2)
= a * cos(B/2) - b * sin(B/2)
cos(A - B/2) = cosA * cos(B/2) + sinA * sin(B/2)
= (cosA + cosB) * cos(B/2) + (sinA + sinB) * sin(B/2)
= b * cos(B/2) + a * sin(B/2)
Now we can substitute the values we found into the expression for tan(A - B/2):
tan(A - B/2) = (sin(A - B/2)) / (cos(A - B/2))
= (a * cos(B/2) - b * sin(B/2)) / (b * cos(B/2) + a * sin(B/2))
And that is the value of tan(A - B/2) in terms of the given values.
Recall your sum-to-product identities:
sinA + sinB = 2 sin(A+B)/2 cos(A-B)/2
cosA + cosB = 2 cos(A+B)/2 cos(A-B)/2
That should get you started.