Discover the Single Solution for 3sinx - 1 = b in (0,2pi): Find the Value of b!

[TyErd]

the equation 3sinx - 1 = b, where b is a positive real number, has one solution in the interval (0,2pi). The value of b is:

Frankly I have no idea where to even start with this problem.

 

[tiny-tim]

Hi TyErd!

(have a pi: π )

Hint: how many solutions in (0,2π) does 3sinx - 1 = 15 have?

And 3sinx - 1 = 0.5 ?

 

[TyErd]

Um. thnx for the pi. How do you know how many solutions there are?

 

[tiny-tim]

Um. thnx for the pi. How do you know how many solutions there are?

Draw a graph and find out.

Get on with it!​

 

[TyErd]

alright I've drawn the graph, so what parts of the graph do i have to look at to know how many solutions there?

 

[TyErd]

hm..

 

[Mentallic]

Which parts? Umm...

If I asked you to find out how many solutions there are to $x^2=10$ how would you go about doing that by looking at a graph?

 

[Mark44]

alright I've drawn the graph, so what parts of the graph do i have to look at to know how many solutions there?

And you should be looking only at the part of the graph on the interval [0, 2π].

 

[tiny-tim]

Nooo … (0,2π).

 

[Mark44]

Nooo … (0,2π).

Right, tiny-tim. I missed that it was the open interval.

 

[TyErd]

so when it says solutions, should i be looking at the x intercepts?

 

[Mark44]

alright I've drawn the graph, so what parts of the graph do i have to look at to know how many solutions there?

What is the graph that you have drawn? Specifically, what is the formula of the function you have graphed?

 

[TyErd]

I tried to draw 3sinx-1=15 by taking 15 onto the other side but I've just realized that isn't right. How do you graph an equation that equals a number?

 

[Mentallic]

Graph $15=3sinx-1$
Then graph $y=15$

Do you see how if you tried to solve these two equations simultaneously, you would get $15=3sinx-1$?

Are there any intersections between the two functions?

Now, can you find any number b (such as 15) that makes it such that the equation $3sinx-1=b$ has only 1 real solution in the interval between 0 and 2π?
This is the same as saying for what number b will the function $y=b$ intersect the function $y=3sinx-1$ only once in the interval $(0,2\pi)$?

 

[TyErd]

Thankyouuu! i get it finally, you made it so much easier. thnx