Approximate the solutions of the given equation in the interval [0, 2 pi)
sin 2x + 1.5 cos x = 0
a. x = 1.624, 1.932, 5.776, 5.997
b. x = 1.055, 3.785, 4.652, 5.721
c. x = 1.101, 2.118, 3.982, 5.104
d. x = 1.484, 13799, 4.626, 5.490
e. x = 1.571, 3.990, 4.712, 5.435
+136 \(\mathrm{Use\:the\:following\:identity}:\quad \sin \left(2x\right)=2\cos \left(x\right)\sin \left(x\right)\)
\(1.5\cos \left(x\right)+2\cos \left(x\right)\sin \left(x\right)=0\)
\(=\cos \left(x\right)\left(2\sin \left(x\right)+1.5\right)\)
\(\cos \left(x\right)=0\:\:\:\mathrm{or}\:\:\:\:2\sin \left(x\right)+1.5=0\)
\(\cos \left(x\right)=0\quad :\quad x=\frac{\pi }{2}+2\pi n,\:x=\frac{3\pi }{2}+2\pi n\)
\(2\sin \left(x\right)+1.5=0\quad :\quad x=2\pi n-0.84806,\:x=2\pi n+\pi +0.84806\)
\(x=2\pi n+\pi +0.84806,\:x=2\pi n-0.84806,\:x=\frac{3\pi }{2}+2\pi n,\:x=\frac{\pi }{2}+2\pi n\)
solve each x between 0 and 2pi
\(0<2\pi n-0.84806<2\pi\)
\(\frac{0.84806}{2pi}<n<1+ \frac{0.84806}{2pi }\)
hence , n = 1 , plug it in the previous equation \(2\pi -0.84806=5.435\)
So without calculating other equation
we have choices
and the true choice is
e. x = 1.571, 3.990, 4.712, 5.435
if you want to complete just solve each other x between 0 and 2 pi and don't forget that n is an integer number
+136 \(\mathrm{Use\:the\:following\:identity}:\quad \sin \left(2x\right)=2\cos \left(x\right)\sin \left(x\right)\)
\(1.5\cos \left(x\right)+2\cos \left(x\right)\sin \left(x\right)=0\)
\(=\cos \left(x\right)\left(2\sin \left(x\right)+1.5\right)\)
\(\cos \left(x\right)=0\:\:\:\mathrm{or}\:\:\:\:2\sin \left(x\right)+1.5=0\)
\(\cos \left(x\right)=0\quad :\quad x=\frac{\pi }{2}+2\pi n,\:x=\frac{3\pi }{2}+2\pi n\)
\(2\sin \left(x\right)+1.5=0\quad :\quad x=2\pi n-0.84806,\:x=2\pi n+\pi +0.84806\)
\(x=2\pi n+\pi +0.84806,\:x=2\pi n-0.84806,\:x=\frac{3\pi }{2}+2\pi n,\:x=\frac{\pi }{2}+2\pi n\)
solve each x between 0 and 2pi
\(0<2\pi n-0.84806<2\pi\)
\(\frac{0.84806}{2pi}<n<1+ \frac{0.84806}{2pi }\)
hence , n = 1 , plug it in the previous equation \(2\pi -0.84806=5.435\)
So without calculating other equation
we have choices
and the true choice is
e. x = 1.571, 3.990, 4.712, 5.435
if you want to complete just solve each other x between 0 and 2 pi and don't forget that n is an integer number
