How do I solve the following question:
sin(5x)cos(3x) - cos(5x)sin(3x) = 1
So far I got this:
sin (8x) = 1
Please help
the identity displayed is ... sin(5x - 3x)
so ... sin(2x) = 1
what do I do next?
To solve the equation sin(5x)cos(3x) - cos(5x)sin(3x) = 1, you correctly simplified it to sin(8x) = 1. Now, to find the solutions for x, you can follow these steps:
Step 1: Take the inverse sine (sin^(-1)) of both sides:
sin^(-1)(sin(8x)) = sin^(-1)(1)
Step 2: Simplify the left side using the inverse and trigonometric properties:
8x = π/2 + 2πn, where n is an integer (due to sin^(-1)(1) = π/2 + 2πn)
Step 3: Divide both sides by 8:
x = (π/2 + 2πn) / 8
So, the general solution for x is x = (π/2 + 2πn) / 8, where n is an integer. This formula provides all the values of x that satisfy the equation sin(5x)cos(3x) - cos(5x)sin(3x) = 1.