Question

At what values in the interval [0,2pi) will the function f(x)=cos 2x +1 and g(x) =sin x +2?

Answer 1

Step 1

Given;

`\begin{gathered} f(x)=cos2x+1 \\ g(x)=sinx+2 \end{gathered}`

Required; To find the intervals within [0, 2pi) the functions will intersect

Step 2

The intervals required are the ones common within the required range to both functions. Graph both functions. The graph shows the points where both graphs intersect within the interval and the coordinates of the points are shown too

or

`\begin{gathered} \sin \left(x\right)+2=\cos \left(2x\right)+1 \\ \sin \left(x\right)-\cos \left(2x\right)+1=0 \\ \sin \left(x\right)+2\sin ^2\left(x\right)=0 \\ \sin \left(x\right)=0,\:\sin \left(x\right)=-(1)/(2) \\ x=0,\:x=\pi,\:\:x=(7\pi)/(6),\:x=(11\pi)/(6) \end{gathered}`

Answers;

`x=0,\pi,(7\pi)/(6),(11\pi)/(6)`