Solve the equation cos ( 2 x ) = √ 2 / 2 on the interval [ 0 , 2 π ) in radians. Separate solutions with a comma. If the equation has no solutions, respond with DNE.

Answer 1

For your trig class, you definitely want to memorize the following six facts about sin and cos for the angles of a circle 30^o which is π/6 in radians, 45^o which is π/4, and 60^o which is π/3:

sin (π/6) = 1 / 2 cos(π/6) = √3 / 2

sin (π/4) = √2 / 2 cos(π/4) = √2 / 2

sin (π/3) = √3 / 2 cos(π/3) = 1 / 2

Then you know that, if cos(2x) = √2 / 2, then 2x = π/4, so x = π/8.

For the other values around the circle, the only other place cos(2x) also equals √2 / 2 is in the IV quadrant at an angle of 360^o - 45^o = 315^o which is 2π - π/4 = 7π/4 radians. So x = 7π/8. (cos is negative in the II and III quadrants.)

The answer is π/8, 7π/8

Answer 2

Hi!

We want to solve cos ( 2 x ) = √ 2 / 2 

From the unit circle, we know that cos (A) = √ 2 / 2  when A = pi/4 (in the first quadrant);

We have also been asked to find all possible values such that cos(A)= √ 2 / 2 ; if we draw the unit circle, we can see that, also 2pi-pi/4 has the same cosine.

Ok! We are ready to answer:

First, let's simplify 2pi-pi/4= 7pi/4.

Then, if we were solving the equation cos (A) = √ 2 / 2 , our answers would be A=pi/4,7pi/4.

But we want to solve cos(2x) = √ 2 / 2 .

It is not a problem. What we do is to set A=2x, now the equation, cos ( 2 x ) = √ 2 / 2  looks like

cos (A) = √ 2 / 2.

We already found

(*) A=pi/4,7pi/4.

But since A=2x, we solve for x => x = A/2. We replace A by (*) =>

x = (pi/4)/2 and x= (7pi/4)/2

We simplify:

x = pi/8, 7pi/8 in [0,2pi]

These are our solutions!

I hope this helps! Please let me know if you have any questions. Good luck.