cos(theta)= 1/2 List 6 specific solutions.

Answer 1

cos(θ) = 1/2 --> θ = cos^-1(1/2) = 60^o

Now, we know cos(θ) = cos(-θ) so cos(-60^o) = 1/2. However, a negative angle is a CW rotation, if we instead do the standard CCW rotation starting at 0^o, we see -60^o is the same angle as 300^o.

That's two solutions. Both of these solutions are less than 360^o, and require less than one rotation around the unit circle. You can get ⋅as many more solutions as you want by adding 360n degrees to each angle. Adding 360 degrees simply rotates around the unit circle exactly once, so 360n is exactly n rotations:

cos(60 + 360n) = 1/2 for n=0,1,2,3,,...

cos(300 + 360n) = 1/2 for n=0,1,2,3,,...

You can let n=0,1,2 in each of the above to get 6 specific solutions:

cos(60^o) = cos(420^o) = cos(780^o) = 1/2

cos(300^o) = cos(660^o) = cos(1020^o) = 1/2

I encourage you to plug in the above values on a calculator to see that they all have cosine of 1/2 (make sure your calculator is set to "degrees" mode). For this problem, you could also just let n=0,1,...,5 in, say, the first equation.