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

Question

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?

Answer 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

Answer 2

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

Answer 3

Page 2 of the following has a nice unit circle,
have it handy

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

Answer 4

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