If A+B=90° then what is the range of SinA•SinB
since A+B = 90
B = 90-A
and sinB = sin(90-A) = cosA
similarly cosB = sinA
sinAsinB = sinAcosA = (1/2)sin 2A
thus y = (1/2)sin 2A
has an amplitude of 1/2
so the range is -1/2 ≤ y ≤ 1/2
To find the range of sinA*sinB when A + B = 90°, we need to determine the possible values of A and B.
Since A + B = 90°, we can rearrange the equation to solve for B:
B = 90° - A
Now, we can substitute this value of B in terms of A into the expression sinA*sinB:
sinA * sinB = sinA * sin(90° - A)
Using the identity sin(90° - A) = cosA, we can simplify the expression:
sinA * sinB = sinA * cosA
Since sinA and cosA are both trigonometric functions of A, we can rewrite the expression as:
sinA * cosA = (1/2)*2*sinA * cosA
Now, we can recognize that (1/2)*2*sinA * cosA is the half-angle formula for sin2A. Therefore, the expression becomes:
sinA * sinB = (1/2)*sin2A
The range of sin2A is between -1 and 1. Therefore, the range of sinA*sinB (or (1/2)*sin2A) is between -1/2 and 1/2.
In conclusion, when A + B = 90°, the range of sinA*sinB is between -1/2 and 1/2.