Understanding the Problem
To express sin(5x) - sin(4x) as a product of trigonometric functions, we can use the sine subtraction formula. This formula helps us rewrite the difference of two sine functions as a product.
Using the Sine Difference Formula
The sine difference formula states:
sin(A) - sin(B) = 2 * cos((A + B)/2) * sin((A - B)/2)
Here, we can identify A and B:
- A = 5x
- B = 4x
Now, we can plug these values into the formula.
Applying the Formula
1. Calculate (A + B)/2:
- (5x + 4x)/2 = 9x/2
2. Calculate (A - B)/2:
- (5x - 4x)/2 = x/2
Now substituting these into the formula:
sin(5x) - sin(4x) = 2 * cos(9x/2) * sin(x/2)
Final Expression
Thus, the expression for sin(5x) - sin(4x) as a product of trigonometric functions is:
- sin(5x) - sin(4x) = 2 * cos(9x/2) * sin(x/2)
Conclusion
This transformation allows us to express the difference of two sine functions as a product, simplifying analysis in various trigonometric applications.