Solve the equations by introducing additional angles: 3sinx-4cosx = 5

Divide the given equation by √ (3 ^ 2 + 4 ^ 2 = 5:

3 / 5sin (x) – 4 / 5cos (x) = 1.

Let’s introduce an additional angle a = arcsin (4/5), then sin (a) = 4/5; cos (a) = 3/5, we get the equation:

sin (x) cos (a) – sin (a) cos (x) = 1.

Using the formula for the sine of the difference of two arguments, we get:

sin (x – a) = 1.

The roots of an equation of the form sin (x) = a are determined by the formula: x = arcsin (a) + – 2 * π * n, where n is a natural number.

x – a = arcsin (1) + – 2 * π * n;

x = π / 2 + arcsin (4/5) + – 2 * π * n.

Answer: x belongs to {π / 2 + arcsin (4/5) + – 2 * π * n}.