if cot (a+b)=0 then write the value of sin(a+2b)
To find the value of sin(a+2b) given that cot(a+b)=0, we can start by rearranging the equation cot(a+b) = 0.
Recall that cot(a+b) is equal to cos(a+b)/sin(a+b). Therefore, when cot(a+b) = 0, it means that the numerator, cos(a+b), is equal to 0, while the denominator, sin(a+b), is not equal to zero.
So, cos(a+b) = 0.
Now, we need to use this information to find sin(a+2b).
We can use the trigonometric identity sin^2(x) + cos^2(x) = 1. Rearranging this identity, we get cos^2(x) = 1 - sin^2(x).
Since cos(a+b) = 0, we substitute this into the identity:
0^2 = 1 - sin^2(a+b).
Simplifying, we find that sin^2(a+b) = 1.
Taking the square root of both sides, we get sin(a+b) = 1 or -1.
Now, we can find sin(a+2b) by using the double angle identity for sine:
sin(2x) = 2sin(x)cos(x).
Applying this identity to sin(a+2b), we have:
sin(a+2b) = 2sin(a+b)cos(b).
Since sin(a+b) can be either 1 or -1, we have two possible values for sin(a+2b):
Value 1: sin(a+b) = 1, which gives us sin(a+2b) = 2 * 1 * cos(b) = 2cos(b).
Value 2: sin(a+b) = -1, which gives us sin(a+2b) = 2 * -1 * cos(b) = -2cos(b).
Therefore, the value of sin(a+2b) when cot(a+b) = 0 can be expressed as either 2cos(b) or -2cos(b), depending on whether sin(a+b) equals 1 or -1.
To find the value of sin(a+2b) given that cot(a+b) = 0, we can use the trigonometric identity:
cot(x) = cos(x) / sin(x)
Since cot(a+b) = 0, we can conclude that cos(a+b) = 0 because sin(a+b) cannot be equal to zero.
Now, let's use the double-angle identity for sine:
sin(2x) = 2sin(x)cos(x)
We can substitute a+2b for x:
sin(a+2b) = 2sin(a+b)cos(b)
Since cos(a+b) = 0, we know that sin(a+b) cannot be zero. So, we can conclude that sin(a+2b) = 0.
Therefore, the value of sin(a+2b) is 0.