Question

Sin x + 22 degrees equals cos 2x - 7 degrees

Answer 1

$\sin(x + 22) = \cos(2x - 7) \\ \sin(x) + \sin(22) \\ = \cos(2x) - \cos(7)$
sin22 = 0.375
cos7 = 0.993
$sin(x) + 0.375 = \cos(2x) - 0.993 \\ sin(x) - \cos(2x) = - .993 - .375 \\ sin(x) - \cos(2x) = -0.368$
cos2x = 1 - 2sin^2x
$sin(x) - (1 - { 2\sin }^{2}x ) = -0.368 \\ {2 \sin }^{2}x + \sin(x) = - .368 + 1 \\ { 2\sin }^{2}x + \sin(x) - 0.632 = 0$
solve like a regular quadratic by completing the square: 2sin^2x + sinx = 0.632
2sin^2x + sinx - 1 = 0.632 - 1
(2sinx - 1)(sinx + 1) = -0.368

2 possible solutions:

1) 2sinx - 1 = -0.368
2sinx - 1 +1 = -0.368+1
2sinx = 0.632
2sinx/2 = 0.632/2
sinx = 0.316
x = sin^-1 (0.316) = 18.42

2) sinx + 1 = -0.368
sinx + 1 -1 = -0.368-1
sinx = -1.368
x = sin^-1 (-1.368) = UNDEFINED

Therefore x = 18.42

Answer 2

No solution, because it woud result in a graph because sin(x+22)=cos(2x-7)  does not possibly equal the same