Question

Consider the equation 3 cos 2x + sin x = 1
(a) Write this equation in the form f(x) = 0, where f(x) = a sin^2 x + b sin x + c
(b) factor f(x)
(c) Write down the number of solutions of f(x) = 0, for 0 < x <2π (do not solve 0)

Answer 1

Use the formula
\[\cos(2 x) = \cos^2(x) - \sin^2(x) = 1- 2 \sin^2(x)
\]

Answer 2

\[3 \cos (2x) + \sin( x)= 1\\
3( 1- 2 \sin^2(x)) + \sin(x) -1=0\\
3 - 6 \sin^2(x) + \sin(x) -1 =0\\
-6 \sin^2(x) + \sin(x) + 2 =0\\
6 \sin^2(x) - \sin(x) - 2 =0\\
(2 \sin (x)+1) (3 \sin (x)-2)=
\]

Answer 3

You should be able to finish it now.