Question

Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.)

Answer 1

sin 2θ + sin θ = 0

Answer 2

sin(2A) = 2sin(A)cos(A)

Use this and factor out sine. Equate each factor to zero and solve for theta.

Answer 3

= 2 sin (theta)cos(theta)

Answer 4

Put it in the original equation, factor out sin(theta)

Answer 5

I am a little lost that gives me 2sin(theta)=0 cos(theta)=0

Answer 6

Let me use x instead of theta for it is easier to type:
sin(2x) + sin(x) = 0
2sin(x)cos(x) + sin(x) = 0
sin(x)(2cos(x) + 1) = 0
sin(x) = 0 or 2cos(x) + 1 = 0
solve for x.

Answer 7

x=0, or x=1/2

Answer 8

so sin=0 or cos =1/2

Answer 9

sin(x) = 0 or cos(x) = -1/2
sin(x) = 0 means x = 0, pi (they want x to be in the interval [0, 2pi)
cos(x) = -1/2 means x = 2/3pi, 4/3pi

So the solutions are: theta = 0, 2/3pi, pi, 4/3pi

Answer 10

is the pi in 2/3 in the denominator with the three or out to the side?

Answer 11

All pi's are in the numerator.

Answer 12

I got it correct thanks so much. I do have another question if you would be willing to help me.

Answer 13

\[\Large \Theta = 0, \frac{ 2 }{ 3 } \pi, \pi, \frac{ 4 }{ 3 } \pi\]

Answer 14

I can assist but you should do the work so you can learn.

Answer 15

alright that is perfectly fine with me. Use an Addition or Subtraction Formula to simplify the equation.

sin θ cos 3θ + cos θ sin 3θ = 0

Answer 16

first I got sin4(theta)= 0
now Im at the second part Find all solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.)
and I don't understand how to go from there

Answer 17

Yes, sin(4theta) = 0 is correct.
Use your calculator or look up a trig table for sine function and see where all it goes to zero in the interval [0, 2π). If using calculator set it to radians.

Answer 18

Okay so what actually am I doing?

Answer 19

its zero at -2pi, -pi, 0, pi, 2pi

Answer 20

find a button labeled arcsin or sin^-1 (sine inverse) amd find the sine inverse for 0.

Answer 21

We are interested only for theta in the interval [0, 2pi) (where 2pi is NOT included because they use a parenthesis ")", but 0 is included in the interval because they use a bracket "["). So 0 and pi are the only values in the interval.

So 4(theta) = 0 or pi
solve for theta.

Answer 22

so theta= 0 or pi/4

Answer 23

Yes.

Answer 24

where I type my answers in online for my homework it says im wrong

Answer 25

Did you enter the answer as a comma separated list. That is: 0, pi/4 ?

Answer 26

yes im thinking there is more then that because in the example that is given it gives a lot of answers. isn't that just the answers for quadrant one don't you have to keep adding until you get through quadrant 4?

Answer 27

Yes that is true because of the 4*theta. So when we look at the arcsin or sine inverse we should not stop at the interval [0, 2pi) but go 4 times as much because of the 4*theta.
So look for arcsin(0) from [0, 8pi) first

Answer 28

We should just keep adding pi:
So it should be (4theta) = 0, pi, 2pi, 3pi, 4pi, 5pi, 6pi, 7pi
so theta = 0, pi/4, pi/2, 3pi/4, pi, 5pi/4, 3pi/2, 7pi/4

Answer 29

alright perfect and I will definitely be asking you for any more help I need. Thank you very much!

Answer 30

you are welcome.