Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2π). (Enter your answers as a comma-separated list. Round your answers to three decimal places . sin 2θ + sin θ = 0

Answer 1

In this problem we have to start with this identity that : sin (2θ)= 2sinθcosθ

1. Sin (2θ) + sin (θ)=0
2. Let substitute Sin (2θ) in the above equation:

2sin(θ)cos (θ) +sin (θ)=0

3. Take out sin (θ) as a common factor I got: Sin(θ) (2 cosθ +1)=0

4 Applying property of zero: Sin(θ)=0. so in the unit circle θ=π, and θ=2θ between 0≤. θ. ≤ 2π

5. Let solve for the other factor. 2cosθ+1=0

subtracting 1 in both sides and dividing by 2 , I got. : cosθ=. -1/2. , looking at the unit circle the answers are θ= 2π/3. and θ= 4π/3.

You final answers : θ=π , 2θ, 2π/3 and 4π/3.

Note: In order to check your answers, graph the giving function in graphing calculator and verify the x-intercepts . ( You must convert the radians values like 2π/3 to decimal values to see the interceptions.

If you still have more questions, please contact me for a great lesson .

Answer 2

Using the Double Angle Formulas we know that sin(2θ)=2sin(θ)cos(θ).

So we can replace sin(2θ) in the initial equation:

sin(2θ) + sin(θ) = 0

2sin(θ)cos(θ) + sin(θ) = 0 Then we can factor out sin(θ)

sin(θ)[ 2cos(θ) +1 ] = 0

Now we can set each factor equal to zero and solve separately

sin(θ) = 0 2cos(θ) + 1 = 0

2cos(θ) = -1

cos(θ)= -1/2

Where does sin(θ) = 0? At θ = 0. Where does cos(θ) = -1/2? At θ = π / 3 and at θ = 4π / 3.

As a list you have θ = 0, π / 3, 4π / 3. If you need decimal places you just need to plug those into a calculator! You can estimate π to be about 3.14159 or you can just use your calculators more accurate value.

Hope that helps!