Question

Please help..medal will be awarded

Answer 1

Solve the equation
\[4\cos2\theta - 14\sin \theta = 7 for 0^{c}\Theta \le2\]

Answer 2

That should be for:
\[0 \le \theta \le 2\Pi \]

Answer 3

\[\cos 2\theta=1-2\sin ^2\theta \]

Answer 4

Make that substitution. Get it equal to 0 and factor.

Answer 5

I tried that..but ended up with a math error on the calculator

Answer 6

\[4(1-\sin ^2x)-14\sin x-7=0\]
\[4-4\sin ^2x-14\sin x-7=0\]
\[-4\sin ^2x-14\sin x-3=0\]
\[4\sin ^2x+14\sin x+3=0\]

Answer 7

That won't factor so plug it into the quadratic formula.

Answer 8

How did you change from a negative to a positive? Did you multiply by -1? If so why?

Answer 9

Because I prefer to have the leading coefficient positive.

Answer 10

And isn't it supposed to be 4(1-2sin^2x)?

Answer 11

yes.

Answer 12

So :
\[8\sin ^2x+14\sin x+3=0\]

Answer 13

Alright

Answer 14

Now it will factor.

Answer 15

I did get it factor, but when I tried to find the principal value..I got an error

Answer 16

\[(2\sin x+3)(4\sin x+1)=0\]
\[\sin x=-\frac{3}{2}; -\frac{1}{4}\]

Answer 17

Discard -3/2 because sin cannot be less than -1

Answer 18

Yes

Answer 19

Do you want the answer in radians or degrees or does it matter?

Answer 20

I ended up with -14.48 for sin=-1/4

Answer 21

I want the answer in radians

Answer 22

That would be degrees.

Answer 23

Set your calculator in radians.

Answer 24

I got -.25 radians but that is not between 0 and 2 pi.

Answer 25

Yes. But since we are looking in the range 0 less than equal to theta less than equal to 2pi...?

Answer 26

So it would be pi+.25 =3.39 or 2pi -.25 or 6.03

Answer 27

How did you get those values?