Trigonometry

Question

Find all solutions to the equation in the interval [0,2pi). Enter the solutions in increasing order.

 

cos 2x = sinx

 

 

Answer 1

We need to get the equation in term of either sine of cosine.  Let's get it in terms of sine.

 

Using the identity     cos(2x) = 1 - 2sin²x

 

 

1 - 2sin²x = sinx

 

 

Make the left side equal to zero so that we have positive coefficient of the sin²x term.

 

0 = 2sin²x + sinx - 1

 

 

Factor using FOIL.

 

0 = (2sinx - 1)(sinx + 1)

 

 

Set the factors equal to zero.

 

2sinx - 1 = 0                and               sinx + 1 = 0

 

 

Solve for x from both equations.

 

sinx = 1/2                                        sinx = -1

 

 

x = π/6                                                 x = 3π/2

x = π - (π/6) = 5π/6

 

 

The three solution are bolded.