Solve this equation. Give a general formula for all the solutions using k to represent your arbitrary integer and using only angles between 0 and 2p.

Question

Solve this equation. Give a general formula for all the solutions using k to represent your arbitrary integer and using only angles between 0 and 2π.

sin(2θ) = -1
θ = ________________+ 2kπ (smaller angle)
θ = ________________ + 2kπ (larger angle)

Answer 1

sin(2θ) = -1

 

Let x = 2θ

 

 

sin(x) = -1

 

x = 3π/2

 

 

Substitute this value of x in terms of θ.

 

x / 2 = θ

 

(3π/2)(1/2) = θ

 

3π/4 = θ

 

 

Since sine is negative in the 3rd and 4th quadrants,

 

θ = -3π/4 + 2kπ , when k = 1   (larger angle)

θ = 3π/4 + 2kπ , when k = 1/2  (smaller angle)

 

 

I hope this helps out alot!