Pre-Cal:(Cont.)

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[Note: I've tried this problem 4 times already and still have it wrong. I know the steps to doing it but for some reason its wrong. an someone please help me get these answers for (b).]

An equation is given. (Enter your answers as a comma-separated list. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)

2 cos 2θ − 1 = 0

(a) Find all solutions of the equation.

My Answer--> θ= (π+6nπ)/6 , (5π+6nπ)/6
(Correct)

(b) Find the solutions in the interval [0, 2π).

θ=____________________________

[Note: my answer was wrong.
π/2,π/6,3π/2,7π/6]

  • Pre-Cal:(Cont.) -

    2 cos 2θ − 1 = 0
    cos 2Ø = 1/2
    I know cos π/3 = 1/2 , and the cosine is positive in I and IV
    so 2Ø = π/3 , in I
    OR
    2Ø = 2π - π/3 = 5π/3 , in IV

    Ø = π/6 or Ø = 5π/6

    (Ø = 30° or Ø = 150°)

    but the period of cos 2Ø is 2π/2 or π
    so adding/subtracting multiples of π to any answer will yield a new answer

    general solution:
    π/6 + kπ or 5π/6 + kπ , where k is an integer

    solutions in [0,2π] :
    π/6, 7π/6, 5π/6, 11π/6

    or

    in degrees: 30°, 210°, 150°, 330°

    verify my answers by using a calculator, they work.

  • Pre-Cal:(Cont.) -

    Thank you so much for your help!!!
    You're so awesome!!! :D

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