Question

HELP PLEASE!!!!!!!

Answer 1

@Mertsj @sourwing @science0229

Answer 2

hello.

Answer 3

hi. can you help me?

Answer 4

Sure.

Answer 5

First, we can realize that the equation is unfactorable, right?

Answer 6

right

Answer 7

For our convenience, I'll say that x=tan(theta)

Answer 8

The equation changes to x^2-x-1=0.
I would suggest that you complete the square to solve this.

Answer 9

If you don't like completing the square, you're always welcome to use the quadratic formula.

Answer 10

ok well x=(1+/-sqrt5)/2

Answer 11

Right. Now, set it equal to tan(theta) since we first assumed that x=tan(theta)

Answer 12

ok i've already done that

Answer 13

Does the problem gives any domain to solve over?

Answer 14

Or do we have to solve it over (-infinity, infinity)

Answer 15

yes \[0\le x \le360\]

Answer 16

ok. Are you more comfortable with degrees or radians?

Answer 17

radians but for this i think we should use degrees just because it says 360 degrees

Answer 18

Ok.

Answer 19

Because there is no "nice" angles that has a tangent value of (1+/-sqrt(5))/2, we have to use inverse tangent.

Answer 20

Let's divide this up into 2 parts
One is that tan(theta)=(1+sqrt(5))/2
Another is that tan(theta)=(1-sqrt(5))/2

Answer 21

Before we start, do you know how to solve simple trig equations, like sin(x)=1/2?

Answer 22

yeah inverse sine arcsin

Answer 23

Ok.
Going back to the original problem, can you solve the first part ?

Answer 24

tan(theta)=(1+sqrt(5))/2
theta=arctan((1+sqrt(5))/2)

Answer 25

Be careful here.
There are 2 solutions.

Answer 26

but this is just the first part. there is only one for each part

Answer 27

Let me draw a graph of y=tanx and y=(1+sqrt(5))/2 to explain why there are 2 solutions.

Answer 28

I can't draw well here, so here is the link to the 2 graphs.

Answer 29

https://www.google.com/search?q=y%3D+tanx%2C+y%3D(1%2Bsqrt(5))%2F2&oq=y%3D+tanx%2C+y%3D(1%2Bsqrt(5))%2F2&aqs=chrome..69i57.176j0j4&sourceid=chrome&espv=210&es_sm=122&ie=UTF-8#q=y%3D+tan+x,+y%3D(1%2Bsqrt(5))/2&spell=1

Answer 30

so what is the answer than? you can't tell with the decimals

Answer 31

Wait.
Here is a perfect image

Answer 32

The first solution is, as you said, arctan((1+sqrt(5))/2)
What do you think is the second solution?

Answer 33

arctan((1-sqrt5)/2)

Answer 34

but is it within [0,360 degrees]?

Answer 35

It is.
If you look at the image, there is a second solution, right?

Answer 36

yeah

Answer 37

I'll give you a hint; the period of tanx is 180 degrees

Answer 38

Because of that, there is a trig identity, stating that\[\tan(x+90)=\tan(x)\]

Answer 39

oooohhh ok yeah it is on the first period so it is within 360. but only 1+sqrt5/2 is,
1-sqrt5/2 is outside 0 right?

Answer 40

Wait. We'll get to (1-sqrt5)/2 later.
Right now, we're only concerned about 1+sqrt5/2

Answer 41

ok what else do we need to do with 1+sqrt5/2

Answer 42

Look at the hint I gave you.
You got an answer of theta=arctan(1+sqrt5/2)
What is the other one?

btw I meant to say \[\tan(90+\theta)=\tan(\theta)\]

Answer 43

not sure. would it just be 90+(1+sqrt5/2)

Answer 44

You're so close.

Answer 45

Was the first answer (1+sqrt5)/2 or arctan((1+sqrt5)/2)

Answer 46

arctan(1+sqrt5/2)

Answer 47

So your second solution will be...

Answer 48

arctan(90+(1+sqrt5/2))

Answer 49

how does 90+arctan((1+sqrt5)/2) sound?

Answer 50

and this is all in degrees?

Answer 51

because theta=arctan((1+sqrt5)/2), and 90+theta is not arctan(90+(1+sqrt5)/2)

Answer 52

Yes.

Answer 53

ok thanks

Answer 54

Oh. Let me give you direction to how to do the second part, which is tan(theta)=(1-sqrt5)/2

Answer 55

isn't it the same thing?

Answer 56

(1-sqrt5)/2 is not the same thing as (1+sqrt5)/2

Answer 57

And, there is a little twist here.

Answer 58

no it doesnt work as a solution 1--sqrt5/2 is negative it doesnt work

Answer 59

It does, and I'll guide you step by step.

Answer 60

but the answer has to be between [0,360] this is below zero

Answer 61

First, look at the 2 graphs, y=tanx and y=(1-sqrt5)/2.
I intentionally drew it over [-pi,2pi]

Answer 62

oh ok

Answer 63

what do i do now?

Answer 64

@science0229?

Answer 65

sorry.
I was eating.

Answer 66

oh sorry ill leave you alone

Answer 67

No it's fine.
I'm done.

Answer 68

OH!
Here are the graphs

Answer 69

ok but would the answer just be arctan((1-sqrt5)/2)? also arctan(90+((1+sqrt5)/2))

Answer 70

No. That's the twist.
arctan((1-sqrt(5))/2) is negative, which is not in the domain.

Answer 71

so it doesnt count as an answer

Answer 72

No. It doesn't.

Answer 73

But your second solution is positive, so it works.
What is the other one?

Answer 74

not sure

Answer 75

Your first solution is 90+arctan((1-sqrt(5))/2)
What do you do to get the second solution?

Answer 76

subtract 90?

Answer 77

opposite.
You're moving 90 degrees to the right.

Answer 78

so 180+(arctan((1-sqrt(5))/2))?

Answer 79

YES!

Answer 80

ok thanks so much

Answer 81

You now have 4 solutions, which are the answers. :)

Answer 82

yep