Final answer:
To find the value of x that satisfies the equation 2cos^8x + sin^4x = 1/27, we can rewrite the equation and factor it to find the values of cosx. From the unit circle, we determine that the solution is x = 30° (Option A).
Step-by-step explanation:
To find the value of x that satisfies the equation 2cos^8x + sin^4x = 1/27, we can first notice that sin^4x can be written as (sin^2x)^2. Then, we can rewrite the equation as 2cos^8x + (sin^2x)^2 = 1/27. We know that sin^2x + cos^2x = 1, so we can substitute (1 - cos^2x) for sin^2x.
This gives us the equation 2cos^8x + (1 - cos^2x)^2 = 1/27. Simplifying further, we get 2cos^8x + 1 - 2cos^2x + cos^4x = 1/27.
By rearranging terms and factoring, we can rewrite the equation as cos^8x - 2cos^2x + cos^4x - 2/27 = 0. This quadratic equation can be factored as (cos^2x - 1/3)^2(cos^4x + 2cos^4x - 2/27) = 0.
From the factored equation, we can see that cos^2x - 1/3 = 0, giving us cos^2x = 1/3. Taking the square root of both sides, we find cosx = ±√(1/3). From the unit circle, we know that cosx = √(1/3) at 30° and cosx = -√(1/3) at 150°. Therefore, the value of x that satisfies the equation is x = 30° (Option A).