Final answer:
The smallest solution for the equation sin(6x)cos(10x) - cos(6x)sin(10x) = -0.7, we apply the sine subtraction identity to rewrite it as sin(-4x) = -0.7, then use the arcsin function and divide by -4 to solve for x.
Step-by-step explanation:
The equation given is a version of the sine subtraction identity, which states that sin(a)cos(b) - cos(a)sin(b) is equivalent to sin(a - b). In your case, the equation can be rewritten as sin(6x - 10x), which simplifies to sin(-4x). Setting this equal to -0.7 and solving for x will give you the smallest solution for the variable x. Remember that sine function values repeat every 2π, so to find the smallest solution, we look within the first period of sine's cycle after considering the negative sign.
Subsequently, you would use the inverse sine function (also known as arcsin) to find the angle whose sine is -0.7. Make sure to properly adjust your angle to be within the appropriate range (since we are using a negative angle for sin(-4x)), and then divide by -4 to solve for x. Bear in mind there will be infinite solutions, but you're asked for the smallest one.