Maths - Please Help?

Question

Hello,

There are some questions that I am not sure how to tackle:

1) sinθ + sin(θ/2) = 0

2) sin(θ/2) = sinθ

The answers are apparently:

1) 0, [2(pi)]/3, 2(pi)

2) 0.253, 2.89

Thank you in advance.

Answer 1

1) sinθ + sin(θ/2) = 0

Using the formula:sinθ=2sinθ/2cosθ/2

2sinθ/2cosθ/2 + sin(θ/2) = 0

sin(θ/2)(2cosθ/2+1)=0

sinθ/2=0    or   2cosθ/2+1=0

sinθ/2=0

sinθ/2=sin0

θ/2=nπ + (-1)\(^{n}\) (0)

θ/2=nπ

θ=2nπ

At n=0     θ =0

At n=1     θ =2π

------------

2cosθ/2+1=0

2cosθ/2=-1

cosθ/2=-1/2

cosθ/2=cos(2π/3)

θ = 2nπ ± ∝

θ/2=2nπ ±2π/3

θ=4nπ ±4π/3

n=0

θ=4π/3

 

 

Answer 2

2) sin(θ/2) = sinθ

Using the formula:sinθ=2sinθ/2cosθ/2

sinθ/2= 2sinθ/2cosθ/2

sinθ/2- 2sinθ/2cosθ/2=0

sinθ/2(1-2cosθ/2)=0

sinθ/2=0

sinθ/2=sin0

θ/2=nπ + (-1)\(^{n}\) (0)

θ/2=nπ

θ=2nπ

2cosθ/2 =1

cosθ/2 =1/2

cosθ/2=cos π/3

θ = 2nπ ± ∝

θ/2=2nπ ± π/3

θ=4nπ ± 2π/3