+136 Subtract 12 from each side
will get \(\sin \left(2x\right)+2\left(2\cos \left(x\right)-3\sin \left(x\right)\right)-12=0\)
Use the identity \(\sin \left(2x\right)=2\cos \left(x\right)\sin \left(x\right)\)
\(\left(2\cos \left(x\right)-3\sin \left(x\right)\right)2+2\cos \left(x\right)\sin \left(x\right)-12=0\)
\(2\left(2\cos \left(x\right)+\cos \left(x\right)\sin \left(x\right)-3\sin \left(x\right)-6\right)\)
\(=2\left(\sin \left(x\right)+2\right)\left(\cos \left(x\right)-3\right)\)
solving each part equivalent to 0
\(\sin \left(x\right)+2=0\:\:\:\mathrm{or}\:\:\:\cos \left(x\right)-3=0\)
sinx cannot be -2
and cos x cannot be 3
hence , no answer in R
Solve for x:
2 (2 cos(x)-3 sin(x))+sin(2 x) = 12
2 (2 cos(x)-3 sin(x))+sin(2 x) = 4 cos(x)-6 sin(x)+sin(2 x):
4 cos(x)-6 sin(x)+sin(2 x) = 12
Subtract 12 from both sides:
-12+4 cos(x)-6 sin(x)+sin(2 x) = 0
Simplify trigonometric functions:
2 (cos(x)-3) (2+sin(x)) = 0
Divide both sides by 2:
(cos(x)-3) (2+sin(x)) = 0
Split into two equations:
cos(x)-3 = 0 or 2+sin(x) = 0
Add 3 to both sides:
cos(x) = 3 or 2+sin(x) = 0
Take the inverse cosine of both sides:
x = cos^(-1)(3)+2 pi n_1 for n_1 element Z or x = 2 pi n_2-cos^(-1)(3) for n_2 element Z
or 2+sin(x) = 0
Subtract 2 from both sides:
x = cos^(-1)(3)+2 pi n_1 for n_1 element Z
or x = 2 pi n_2-cos^(-1)(3) for n_2 element Z
or sin(x) = -2
Take the inverse sine of both sides:
Answer: | x = cos^(-1)(3)+2 pi n_1 for n_1 element Z
or x = 2 pi n_2-cos^(-1)(3) for n_2 element Z
or x = pi+sin^(-1)(2)+2 pi n_3 for n_3 element Z or x = 2 pi n_4- sin^(-1)(2) for n_4 element Z
