Question

How many solutions exist for the equation cos 2θ − sin2θ = 0 on the interval [0, 360°)?

5
4
3
2

Answer 1

I need some help

Answer 2

\[\cos 2\theta-\sin2 \theta=0,\cos 2\theta=\sin 2\theta\]
\[\tan 2\theta=1=\tan 45,\tan (180+45)\]
\[=\tan45,\tan 225\]
\[=\tan \left( 45+180n \right),\tan (225+180n),n=0,1\]
\[2\theta=45+180n,225+180n,n=0,1\]
when n=0
\[2\theta=45+180\times 0,225+180\times 0\]
\[2\theta=45,225\]
\[\theta=\frac{ 45 }{ 2 },\frac{ 225 }{ 2 }=22.5,112.5\]
when n=1
\[2\theta=45+180\times1,225+180\times 1\]
\[2\theta=225,405\]
\[\theta=112.5,202.5\]
when n=2
\[2\theta=45+180\times2,225+180\times 2\]
\[2\theta=45+360,225+360\]
\[2\theta =405,585\]
\[\theta=202.5,292.5\]
when n=3
\[2\theta=45+180\times3,225+180\times 3\]
\[2\theta=45+540,225+540\]
\[2\theta=585,765\]
\[\theta=292.5,382.5\]
\[\theta=22.5,112.5,202.5,292.5\]
so there are four solutions.

Answer 3

Okay wow this makes so much sense great job, thank you so much!

Answer 4

n=0,1,2,3