Question

(tan^2 θ − 16)(2 cos θ + 1) = 0

Answer 1

Have you considered factoring the difference of squares on the tangent piece? Then you're off!

Answer 2

how do you do that?

Answer 3

Factor this expression: x^2 - y^2

Answer 4

(x + y)(x - y)

Answer 5

Perfect. Do the exact same thing with this expression:

(tan^2 θ − 16)

Answer 6

\[(\tan \theta +4)(\tan \theta-4)\]

Answer 7

Super.

\((tan(\theta) − 4)(tan(\theta) + 4)(2 cos θ + 1) = 0\)

Now, what do you do with a fully factored expression set to zero to find all the possible solutions?

Answer 8

so \[\tan \theta+4=0? \tan \theta-4=0? 2\cos+1=0?\]

Answer 9

You have it!! Track down those solutions. There are lots.

Answer 10

i hsvr twenty of these if youre willing to work these out with me

Answer 11

Did you get this one? I haven't seen the results, yet.

Answer 12

1.325 -1.325 pi/3

Answer 13

are those all of the solutions?

Answer 14

Yes, plus many more, since tangent and cosine are periodic.

2cos(x) + 1 = 0

cos(x) = -1/2

x = 2pi/3 + 2kpi AND 4pi/3 + 2kpi for any integer, k. That's quite a few solutions - infinitely many!

Do the same thing with the tangents.

Answer 15

Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) those are the contraints added to the problem

Answer 16

There you go. I did it for the cosine piece. You have an initial answer for two tangent pieces. You will need to find the rest.

Answer 17

Tangent is positive in 1st and 3rd quadrants.

Answer 18

so its x= pi/3+kpi and 5pi/3+kpi??

Answer 19

what my problem is that im not understanding the fundamental concepts. I need something to explain how each of these work

Answer 20

No, cos(x) = -1/2

x = 2pi/3 or 4pi/3

You're in the wrong quadrants.

Answer 21

ohh I was in quadrents 1 and 4


so are those the only two for cos?



so that makes the tan answers: only 1.325 and -1.325

Answer 22

No, there are lots more to go.

You have x = 1.325. What is the period of the tangent function?

Answer 23

im guessing its 2pi
it doesnt say

Answer 24

Good guess, but it's just pi. You're almost done!

Answer 25

so 1.325, and 2pi/3 4pi/3

Answer 26

No, where did the k go and where did all the other infinitely many solutions go?

\(\theta = 2\pi/3 + 2k\pi\)
\(\theta = 4\pi/3 + 2k\pi\)
\(\theta = 1.325 + k\pi\)
\(\theta = -1.325 + k\pi\)

That's a whole lot of answers!