Final answer:
To solve the equation sin(4x)cos(6x)−cos(4x)sin(6x)=−0.9 for the smallest positive solution, we can use the trigonometric identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B). By applying this identity and performing some calculations, we find that x ≈ -148.3 degrees (or -148.3 * π / 180 radians).
Step-by-step explanation:
To solve the equation sin(4x)cos(6x)−cos(4x)sin(6x)=−0.9, we can use the trigonometric identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B). By applying this identity, the equation can be rewritten as sin(4x - 6x) = -0.9. Simplifying further, we get sin(-2x) = -0.9. To find the smallest positive solution, we can take the inverse sine (sin⁻¹) of both sides of the equation.
By using a calculator, we find that the inverse sine of -0.9 is approximately -63.4 degrees. Since we want the smallest positive solution, we need to add 360 degrees to obtain the positive angle. So, -63.4 + 360 = 296.6 degrees. Finally, we divide the angle by the coefficient of x to find the value of x. Therefore, x ≈ 296.6 / -2 = -148.3 degrees (or -148.3 * π / 180 radians).