Question

cos2theta = 3sintheta + 2

Answer 1

First, we make the problem easier by setting:
Why? because this puts us in a linear space; linear is easy.
sin theta = x
cos theta = y
which gives
3x-4y=2
or
y=3/4x-1/2 (eq. 1)
if we remember our trig identities
now we also know sin^2 + cos^2 =1 so
x^2+y^2=1 (eq. 2)

so now that we have two independent equations for x and y, we can solve for them by reduction
replace y in eq 2 with eq 1 and solve for x. (this is a nasty one.)
x^2 + (3/4x -1/2)^2 = 1
x^2 + 9/16x^2 - 6/8x +1/4 = 1
25/16x^2-6/8x-3/4=0
25x^2-12x-12=0
use the quadratic equation to yield
x=-0.493
OR
x=0.973

since x= sin theta we plug in our two x's (one at a time) to reveal:

theta = 76.7 degrees or -29.6 degrees
why two answers?
since sin theta, cos theta sweep a circle, a linear relationship can have 0, 1, or 2 solutions.

Answer 2

I hope this helps hun ^^^

Answer 3

hun, huh? Are you from Baltimore of the south? (Just wondering)

Answer 4

*or the

Answer 5

Larry i put something above it will explain xD

Answer 6

I know the answer, I was asking you personally :)

Answer 7

\(\cos(2\theta) = 3\sin(\theta) + 2\)

\(1 - 2\sin^{2}(\theta) = 3\sin(\theta) + 2\)

\(2\sin^{2}(\theta) + 3\sin(\theta) + 1 = 0\)

\((2\sin(\theta) + 1)(\sin(\theta) + 1) = 0\)

\(\sin(\theta) = -1/2\;or\;\sin(\theta) = -1\)

You're almost done.

Of course, if you meant \(\cos^{2}(\theta)\), then that's a horse of a different color.