Find the solutions in [0,2π) to the nearest hundredth:

9cos x+5=0

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 [0,2π), the sinx=0 for x=0 and x=π=3.1416→x=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=1/2→x=cos^(-1) (1/2)→x=π/3=3.1416/3→x=1.05

Fourth quadrant: x=2π-π/3=(6π-π)/3→x=5π/3=5(3.1416)/3→x=5.24

Solutions: x=0, 1.05, 3.14, and 5.24

Step-by-step explanation:

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