solve: 7sin^2 theta -22 sin theta +3 =0

Mathematics
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7sin^2 theta -22 sin theta +3 =0

separate answers by commas

May 2nd, 2015

Hi! So we have what looks like a quadratic here.  What you are going to do is substitute some variabe in for sin theta.  So let's pick y.

We have:   7y^2-22y+3=0

Let's apply the quadratic formula to find y.

y=(22+-(22^2-4(7)(3))^1/2)/14

After our calculations we find:

y=1/7 and y=3

Now, we plug sin theta back in for y.

sin theta=1/7

sin theta=3

Then we take arcsin of both sides to get theta. 

theta=0.1433475689053653575950742801566183260989392131334643 

So 3 is a special case where we get an imaginary number which can be expressed two ways.

theta= 

1.57079632679489661923132169163975144209858469968755291048747... -
1.76274717403908605046521864995958461805632065652327082150659... i
(result in radians)
OR

r = 2.36108 (radius), theta = -48.2955° (angle)
Please let me know if you have any questions!  Hope this helps!

May 2nd, 2015

ok so second approach, let's factor it

we get 

(y-3) (7 y-1) = 0
So sin theta-3=0 or 7sin theta-1=0

May 2nd, 2015

Let's throw out the imaginary case, sin theta=3

so we get that sin theta=1/7 again, but this time it is a little neater.  Now, we ned to determine when that happens.  We know we have a 1, sqrt(48), 7 triangle. Which is not pretty. so, now how to make it any nicer.... hmmm


May 2nd, 2015

Oh we get to round! that is convenient!  So theta=0.143 +2kpi

That gives us every revolution.  So now we need to see if any other angles can have a sine of 1/7

May 2nd, 2015

So sine is positive in the first quad and second. We now need the value of that second quadrant angle. So we get it by doing (pi-0.143)  So our second answer is 

theta= 

2.999+2kpi

May 2nd, 2015

not sure why it's not working

May 2nd, 2015

My best guess is because you only wen to two decimal places on the 2.999


But other than that, I honestly have no clue what they would be picking at.

May 2nd, 2015

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May 2nd, 2015
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May 2nd, 2015
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