Solve sin ( 5 x ) cos ( 10 x ) − cos ( 5 x ) sin ( 10 x ) = − 0.45 for the smallest positive solution.

Answer 1

x = 5.3 degrees

You can simplify the problem. Let y=5x, then

sinycos2y - cosysin2y = -0.45

use the formula sin(A-B) = sinAcosB - cosAsinB to get

sin(y-2y) = -0.45

sin(-y) = -0.45

-siny = -0.45

siny = 0.45

y = sin^-1(0.45) = 26.74

y=5x

x = 26.74/5 = 5.3 degrees

check the answer

sin(206.74 - 413.48) = sin(-206.74) = - sin(206.74) =- sin(26.74) = 0.45

Answer 2

sin(5x)cos(10x) - cos(5x)sin(10x) = -0.45

sin(5x)cos(5x + 5x) - cos(5x)sin(5x + 5x) = -0.45

Using the angle addition identities:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y) and cos(x + y) = cos(x)cos(y) – sin(x)sin(y)

gives:

sin(5x)[cos(5x)cos(5x) - sin(5x)sin(5x)] - cos(5x)[sin(5x)cos(5x) + cos(5x)sin(5x)] = -0.45

sin(5x)[cos²(5x) - sin²(5x)] - cos(5x)[2sin(5x)cos(5x)] = -0.45

sin(5x)cos²(5x) - sin³(5x) - 2sin(5x)cos²(5x) = -0.45 [combining like terms gives us:

-sin(5x)cos²(5x) - sin³(5x) = -0.45

Factoring out a -sin(5x) gives us:

-sin(5x)[cos²(5x) + sin²(5x)] = -0.45 [using the Pythagorean Identity gives us:

-sin(5x)(1) = -0.45

sin(5x) = 0.45

Let w = 5x

sin(w) = 0.45

w = sin^-1(0.45)

w = 0.466765 (smallest answer)

Back substituting: 5x = 0.466765

x = 0.466765/5 = 0.09335