Trigonometry

Find all the angles θ in the interval [-2pi, 2pi] for which cos(3θ) = 1/sqrt(2)

I'm not sure why I would have to look for angles in 3 revolutions of the circle (3 positive, 3 negative).

Also how would I convert cos(3θ) = 1/sqrt(2) to a radian value? Do I just need to know the unit circle by heart?

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  1. You should know, or by use of your calculator, that cos(±45°) = 1/√2
    so 3θ = ±45° and θ = ±15°
    now the period of cos(3θ) is 360°/3 or 120°
    so by repeatedly adding and subtracting 120° to any answer will yield a new answer
    Your domain is -360° to 360°, so answers in degrees are:
    ±15, ±135, ±255, ±105, ±225, and ±345

    in radians, you know that π/6 = 30°
    so 15° = π/12
    to quickly convert the above degrees to radians I do the following steps
    135/15 = 9
    then 135° = 9(π/12) = 3π/4
    continuing in this way, the radian answers are:
    ±π/12, ±3π/4, ±17π/12, ±7π/12, ±5π/4, and ±23π/12

    Proof:
    www.wolframalpha.com/input/?i=plot+y+%3D+cos(3x)+,+y+%3D+1%2F%E2%88%9A2+from+-2%CF%80+to+2%CF%80

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  2. The above post was my me, forgot to put in my name

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    posted by Reiny
  3. Page 2 of the following has a nice unit circle,
    have it handy

    tutorial.math.lamar.edu/pdf/trig_cheat_sheet_reduced.pdf

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    posted by Reiny
  4. Thanks. I'm not permitted to use a calculator, which makes this a lot harder.

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    posted by Pat

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