Question

Solve the equation on the interval [0, 2π).

tan2x sin x = tan2x

Answer 1

Have you considered factoring?

tan2x sin x - tan2x = 0

[tan2x][sin x - 1] = 0

Answer 2

I don't know how to factor.

Answer 3

You never took Algebra I? or Algebra II?

Answer 4

I have, but this is trig.

Answer 5

You will need ALL your algebra. Perhaps you should take some time to review or you will have a miserable time with trigonometry.

Answer 6

*having a miserable time

Answer 7

Okay, how long ago was Algebra II?

Answer 8

Not long, but I hardly retained any of it.

Answer 9

I hardly passed the class.

Answer 10

Not good. Do you recall the Distributive Property?

Answer 11

Somewhat

Answer 12

That's all we need for this one.

We had tan(2x)sin(x) = tan(2x)

First, I just subtracted tan(2x) from both sides.

tan(2x)sin(x) - tan(2x) = 0

See how that worked?

Answer 13

yes

Answer 14

Here's the Distributive Property

a(b+c) = ab + ac

Or the other way

ab + ac = a(b+c)


Ringing any bells?

Answer 15

That I'm familiar with.

Answer 16

Perfect. That's all that happened.

We had this:

tan(2x)sin(x) - tan(2x) = 0

We used that Distributive Property to achieve:

tan(2x)(sin(x) - 1) = 0

In this case

a = tan(2x)
b = sin(x)
c = -1

Do you see all those pieces?

Answer 17

Makes a lot more sense now.

Answer 18

Well, that's just the algebra - FACTORING! Ready for the trigonometry?

Answer 19

I don't think I have much of a choice.

Answer 20

:-)

Okay, the point of factoring lies in an odd property of zero.

If I have two numbers, and they are multiplied, giving zero (0), what can I say about the numbers?

If a*b = 0, what is there to know about 'a' or 'b'?

Answer 21

They equal zero?

Answer 22

Not necessarily BOTH, but one of them MUST! Do you see the difference?

3*0 = 0

0*4 = 0

0*0 = 0

Answer 23

Yes

Answer 24

Okay, now we're back here.

tan(2x)(sin(x) - 1) = 0

We have tan(2x) multiplied by (sin(x) - 1). They are multiplied and we get zero.

Therefore, either

tan(2x) = 0

or

sin(x) - 1 = 0

Agreed?

Answer 25

Agreed

Answer 26

Let's tackle them backwards

If sin(x) - 1 = 0, then sin(x) = 1.

Where on [0,2pi) is the sine function equal to +1? It's your first trig question of the day! It's been all algebra to this point.

Answer 27

lost

Answer 28

Do you have a Unit Circle? It's important.

Answer 29

No

Answer 30

Then we have to stop here. You'll need to gain this additional information or we cannot proceed. I'm guessing you have a book with it in there. Here's a link with one.

http://www.thattutorguy.com/unit-circle-pdf/?gclid=CNe_y6OnkLkCFelAMgodXhUAYA

There are lots of others.

Points around the unit circle have coordinates attached to them. The one at the top is (0,1). This means the cosine is 0 and the sine is 1 for that point.

Well, good luck. This is where you are stuck. You'll have to get this circle stuck in your head and come on back.

Answer 31

Holy crap

Answer 32

Alright, I'll work on it. Thanks for your help.

Answer 33

It is a lot at first glance.
Start with the horizontal and vertical. Get those four in your head.
Then cut those in half and work on the NE, SE, SW, and NE corners.
It will start to soak in.