Use a calculator to solve the equation on the interval [0, 2Ο). Round to the nearest hundredth of a radian. sin 2x - sin x = 0 : 0, 1.05, 3.14, 5.24 1.05, 3.14, 5.24 0, 2.09, 4.19 0, 2.09, 3.14, 4.19
Accepted Solution
A:
sin 2x - sin x=0 Using the trigonometric identity: sin 2x=2 sinx cosx 2 sinx cosx - sinx =0 Common factor sinx sinx ( 2 cosx -1)=0 Two options: 1) sinx=0 on the interval [0,2Ο), the sinx=0 for x=0 and x=Ο=3.1416βx=3.14
2) 2 cosx - 1=0 Solving for cosx 2 cosx-1+1=0+1 2 cosx = 1 Dividing by 2 both sides of the equation: (2 cosx)/2=1/2 cosx=1/2
cosx is positive in first and fourth quadrant: First quadrant cosx=1/2βx=cos^(-1) (1/2)βx=Ο/3=3.1416/3βx=1.05 Fourth quadrant: x=2Ο-Ο/3=(6Ο-Ο)/3βx=5Ο/3=5(3.1416)/3βx=5.24