trigonometry graphs/ratios question

[hearts123]

Hey everyone, I'm having some trouble with this homework question. Any help would be appreciated!

I was given this periodic function: f(x) = -f(-x)
The period, T=5
f(-1) = 1
tan θ = 3
What is the value of: f(20sinθcosθ) ?

I tried to use the inverse tan function to find θ, and using that angle value I found what 20 * sin θ * cos θ would be equal to. However, I don't know how to continue from there, and I'm not sure how to incorporate the T=5, nor the rest of the given information.

p.s. I'm not too sure which category this question should go in, since it has both trigonometry and functions? So I guessed and posted it here

 

[Dr.Peterson]

The information given will only be sufficient if the numbers are just right, so I could guess approximately what will happen. And it does.

What did you get for 20 sin θ cos θ? That will be the key to the rest of the work, so we need to see that! Once you write the required answer as f(__), you can use the period to express it in terms of the one known value, f(-1). It will also be helpful to take a moment to think about what the fact that the function is odd, i.e. f(x) = -f(-x), implies in connection with the given value.

You don't really have to use the inverse tangent; you can use identities, or a right triangle, to determine what sinθ and cosθ are, and be sure of an exact answer.

By the way, this definitely does belong under trig, both for the trig functions involved and the importance of periodicity. But it doesn't really matter much, as anyone able to help will see it anyway.

 

[hearts123]

The information given will only be sufficient if the numbers are just right, so I could guess approximately what will happen. And it does.

What did you get for 20 sin θ cos θ? That will be the key to the rest of the work, so we need to see that! Once you write the required answer as f(__), you can use the period and the oddness of the function to express it in terms of the one known value, f(-1).

You don't really have to use the inverse tangent; you can use identities, or a right triangle, to determine what sinθ and cosθ are, and be sure of an exact answer.

By the way, this definitely does belong under trig, both for the trig functions involved and the importance of periodicity. But it doesn't really matter much, as anyone able to help will see it anyway.

Hey, thanks so much for the reply! The value for 20 sin θ cos θ was 6. (θ was about 71.56)
I'm guessing that since it turned out to be a whole number (6) and not a random decimal, what you said is correct; the numbers have to be just right.
Also, I'm not too sure what you mean by "you can use the period and the oddness of the function to express it in terms of the one known value, f(-1)"
Would it be okay for you to explain and walk me through doing that?

 

[Dr.Peterson]

Okay. First, given that f(-1) = 1 and f(x) = -f(-x), what other function value do you know?

Then, compare the fact that your function input will be 6 with the two inputs for which you now know the function output. Does that work nicely with the periodicity, i.e. the fact that f(x + 5) = f(x) for any x?

As for the exactness of the 6, if you make a right triangle with opposite = 3 and adjacent = 1, you can find exact values for sin θ and cos θ and show that 20 sin θ cos θ is exactly 6. If it weren't exact, you really couldn't give an answer at all.

 

[hearts123]

I know that visually, odd functions are symmetrical. So is the other function value I know, f(1) = -1 ?
That way, if I use x=1, I'll get f(1+5) = f(1).
So, f(6) equals to the same value that f(1) is equal to.
So f(6) = -1?

 

[Dr.Peterson]

I think you've got it.

Since f(x) = -f(-x), and we know that f(-1) = 1, we can let x = 1 and see that f(1) = -f(-1) = -1.

Since f(x + 5) = f(x) and we know that f(1) = -1, we can again let x = 1 and see that f(1 + 5) = f(1), that is, f(6) = -1.

Visually, we are taking the point (-1, 1) on the graph and reflecting it to (1, -1), and then using periodicity to move that to (6, -1).