Question

Help with Solving Trigonometric Equations Involving Multiple Angles

Answer 1

How many solutions does the following equation have on the interval [0, 2π]?

\[4\cos(2\Theta)=8\cos^2(2\Theta)\]

Answer 2

\[\cos(2\Theta)=4\cos^2(2\Theta)\]

Answer 3

@phi

Answer 4

write it as
\[ 4\cos^2(2\Theta)- \cos(2\Theta)= 0 \\ \cos(2\Theta) ( 4 \cos(2\Theta) -1)=0
\]
solve
\[ \cos(2\Theta) = 0 \\ \text { and} \\4 \cos(2\Theta) = 1 \]

Answer 5

\[\cos(\cos(2\Theta))=0 = \Theta=\cos2\]

Answer 6

cos^-1(2)**

Answer 7

although I notice if you start with
\[ 4\cos(2\Theta)=8\cos^2(2\Theta) \]
that simpifies to
\[ \cos(2\Theta)=2\cos^2(2\Theta) \]
so you should solve for
\[ \cos(2\Theta )=0 \]
and
\[ \cos(2\Theta)= \frac{1}{2} \]

Answer 8

question did I do the first equation right so far?

Answer 9

I would sketch the cosine curve