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Solve the following.  Find all solutions from 0 degrees to 360 degrees.

 

1.  4 sin^2 x – 1 = 0

 

2.  sin^2 x + sin x = 1

 Apr 17, 2019
 #1
avatar+106519 
+1

1.  4 sin^2 x – 1 = 0    add 1 to both sides

 

4sin^2x  = 1       divide both sides by 4

 

sin^2 x   = 1/4          take both roots and we have that

 

sin x = √[1/2 ]  and this happens at  45 degrees and  135 degrees    

 

And

 

sin x = - √[1/2]   and this happens at   225 degrees and 315 degrees

 

 

cool cool cool

 Apr 17, 2019
edited by CPhill  Apr 17, 2019
 #2
avatar+106519 
+1

2. sin^2 x + sinx  = 1

 

sin^2 x + sin x - 1  =  0

 

Let a = sin x

 

a^2 + a - 1  = 0

a^2 + a= 1       complete the square on a

a^2 + a + 1/4  = 1 + 1/4

(a + 1/2)^2  =  5/4

 

Take both roots  

 

a + 1/2 = √5/2

a =  √5 - 1

        _____

           2

 

sin x = √5 - 1

            _____    take the arcsin and x  ≈ 38.17°   and (180 - 38.17) ≈ 141.8°

               2

 

Or

 

a  =   -√5 + 1

        ______      

             2

 

sin (x)  =  - √5  - 1

                _______      this answer is < -1....so no answer for sin x  

                      2

 

 

cool cool cool

 Apr 17, 2019

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