Find the sum of the three smallest positive values of theta such that 4 cos^2(2theta-pi) =3. (Give your answer in radians.)

Hi guys, I've been struggling with this problem. Here's my thinking about the problem:
we can rewrite the equation as cos^2(2theta-pi) = 3/4
Then we sqrt both sides --> cos(2theta - pi) = sqrt3 / 2
we find what arccos sqrt3 / 2 ---> pi/6 radians or 45 degrees (we'll use radians)
thus cos (2theta - pi) = cos(pi/6)
so 2 theta - pi = pi/6
and 2 theta = 7pi/6 ---> 7pi/12
When we let arccos sqrt3/2 ---> 11pi/6 and 13pi/6 we get 17pi/12 and 19pi/12 (those are the smallest two values which are positive, meaning 11pi/6 and 13pi/6)
so in all we get 17pi/12 + 19pi/12 + 7pi/12 = 43pi/12
Is this right?

Hmmm.

4cos^2(2θ-pi) = 3
cos^2(2θ-pi) = 3/4

since cos(2θ-pi) - -cos2θ, this makes things simpler, so we have

cos^2 2θ = 3/4
cos2θ = ±√3/2
2θ = π/6,5π/6,7π/6,...
θ = π/12,5π/12,7π/12
Looks like the sum is 13π/12

Oh thanks that helped a lot :)

if sinA= 1/3, then find sin (A+pi/6), cos(A-pi/3), tan(A-pi4)

Your process and final answer are almost correct, but there's a small mistake in one of your calculations.

You correctly state that cos(2theta - pi) = sqrt(3)/2. To find the possible values of theta, you take the inverse cosine of both sides: arccos(sqrt(3)/2) = π/6 or 11π/6.

You then solve the equation 2theta - pi = π/6 and get 2theta = 7π/6, so theta = 7π/12.

Next, you solve 2theta - pi = 11π/6 and get 2theta = 13π/6, so theta = 13π/12.

At this point, you made a mistake when you calculated the values of theta. The correct calculations should be:

For theta = 7π/12, the value is positive and one of the smallest values.

For theta = 13π/12, the value is negative and therefore not one of the smallest values.

So, the sum of the three smallest positive values of theta is:

(7π/12) + θ₁ + θ₂

To find θ₁ and θ₂, you need to add 2π to each of the valid theta values you found.

Adding 2π to 7π/12, we get 7π/12 + 24π/12 = 31π/12.

Adding 2π to 11π/6, we get 11π/6 + 12π/6 = 23π/6.

Now, the sum of the three smallest positive values of theta is:

(7π/12) + (31π/12) + (23π/6) = (60π/12) = 5π.

So, the final answer is 5π radians.