Final answer:
The smallest possible solution to the equation sin(4x)cos(9x)-cos(4x)sin(9x)=-.45 is x = -0.132.
Step-by-step explanation:
To find the smallest possible solution to the equation sin(4x)cos(9x)-cos(4x)sin(9x)=-.45, we can use trigonometric identities. We know that sin(a-b) = sin(a)cos(b) - cos(a)sin(b), so we can rewrite the equation as sin(4x-9x) = -.45. Simplifying, we get sin(-5x) = -.45. To find the smallest possible solution, we need to determine the value of x that makes sin(-5x) equal to -.45.
Since the range of the sine function is between -1 and 1, and we are looking for a negative value, we can use the arcsine function to solve for x. Taking the arcsine of -.45, we get approximately x = -0.132. Therefore, the smallest possible solution to the equation is x = -0.132.