Question

Sin5x*cos2x+cos5x*sin2x=1

Answer 1

Sin5x*cos2x + cos5x*sin2x = sin(5x + 2x) = sin7x

Answer 2

To solve the equation sin(5x) * cos(2x) + cos(5x) * sin(2x) = 1, we can use the trigonometric identity for the sum of angles:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b).

Comparing this identity with the given equation, we can see that a = 5x and b = 2x.

Using the identity, the equation becomes:

sin(5x + 2x) = 1.

Simplifying, we get:

sin(7x) = 1.

To find the value(s) of x that satisfy this equation, we need to determine when sin(7x) equals 1. Since the range of the sine function is -1 to 1, the equation sin(7x) = 1 has solutions when 7x is an odd multiple of π/2. In other words:

7x = (2n + 1) * π/2,

where n is an integer.

Now, solve for x:

x = (2n + 1) * π/14.

This equation gives the general solution for x. By substituting different integer values for n, you can find multiple values of x that satisfy the equation sin(5x) * cos(2x) + cos(5x) * sin(2x) = 1.