solving trig function?

Question

I need to give a general formula and six solutions.

sin (theta)=sqrt (2)/2

Heres what i have so far:

theta= pi/4+2 pi k or 3pi/4 + 2 pi k, k being any integer

I believe that is the right general formula for finding the solutions.

theta= pi/4, 3pi/4.......

I need help finding the other 4 solutions. not sure if i use a unit circle or something else

thanks for any help offered

Answer 1

You've already answered your own question.

Between 0 and 2π, there are only two solutions for sin(θ)=√2/2, and those are π/4 and 3π/4. All other possible solutions are additives of 2πk, k any integer, as you've already determined.

There are an infinite number of solutions, but you only need six. You've found two. To find four more, simply substitute real integers for k, e.g., k=1 and k= -1:

k=1: θ=9π/4,11π/4;
k= -1: θ= -7π/4, -5π/4.

Good luck, work hard, and stay away from drugs.

Answer 2

Okay you ready. You are on the right track. sin(theta)=rt2/2, take sin and move it to the right of the equal sign by doing arcsin(rt2/2), on the unit circle sin represents all the 'y' values right, so any answers for positive root2/2 are in Quad 1&2, using the unit circle we find pi/4 and its counterpart 3pi/4(180deg [or pi] -pi/4), then we must apply all the solutions for sin between 0pi and 2 pi (neg x-axis and pos x-axis or all the positive 'y' values, Quad 3&4 are negative sin values) by adding 2pi(k) to both pi/4 and 3pi/4. Starting with k=-1,0,1.
pi/4+2pi(-1) = pi/4-2pi or -7pi/4, 3pi/4+2pi(-1)= -5pi/4
pi/4+2pi(0)=pi/4, and k=0 for 3pi/4 +2pi(0)=3pi/4, for k=1 we have answers beyond our domain reqs of 2pi with 9pi/4 and 11pi/4 rerspectively. Therefore, our solution set is only pi/4 and 3pi/4 because they fall into our respective domain of 0pi and 2pi.
Hope this helps out.

Answer 3

wait you are making this way too complicated try this. First find exact number for sqrt (2)/2 then find the anti-sine (shoudl be on same button as sin just have to go to second function of that button) of that.

Answer 4

what made you think there should be six solutions?

draw a graph of sin(x) [just do a wave, it doesn't need accuracy

draw any line horizontal to x axis. its either above/below the wave and no solutions, or its x=1 or x=-1 and has one solution, or for |x|<1 there are two solutions where your line cuts the wave.

Answer 5

sin Ø = √2 / 2
Ø is in quadrants 1,2
Ø = π / 4 , 3π / 4
Further answers are obtained by adding 2π to these values as required:-
Ø = π / 4, 3π / 4
Ø = 9.π / 4 , 11.π / 4
Ø = 17.π / 4, 19.π / 4