Solve the equation sin(2t) + cost= ¼ on the interval [0, 2π]. Express answer(s) to three decimal places.

Question

anonymous · Answer

I graphed y=sin(2t)+cost and then y=1/4, but they are coming out as straight lines. did I do something wrong?

anonymous · Answer

$$
\sin(2t)+\cos(t)={1\over4}\
2\sin(t)\cos(t)+\cos(t)={1\over4}\
\cos(t)\left[2\sin(t)+1\right]={1\over4}\
\cos(t)={1\over 4\left[2\sin(t)+1\right]}\
\;\
\text{we have the relation}\
\sin^2(t)+\cos^2(t)=1\
\sin^2(t)+{1\over16\left[2\sin(t)+1\right]^2}=1
$$
solve the last quadratic equation for t