# Physics textbook, replace sine with its definition (?)

1. Jul 23, 2012

### 1MileCrash

physics textbook, replace sine with its definition (???)

What on earth do they mean?

"That will introduce either sin(theta) or cos(theta). Reduce the resulting two variables, x and theta, to one, x, by replacing the trigonometric function with an expression (its definition) involving x and y."

Replace sin(theta) with a function of x and y? What?

2. Jul 23, 2012

### Muphrid

Re: physics textbook, replace sine with its definition (???)

Sounds like this is on a 2d plane, in which case $\theta$ denotes the counterclockwise angle from the x-axis, and any trig function of $\theta$ can be written in terms of x and y.

3. Jul 23, 2012

### 1MileCrash

Re: physics textbook, replace sine with its definition (???)

Both? Or one or the other?

4. Jul 23, 2012

### DrGreg

Re: physics textbook, replace sine with its definition (???)

Unless you tell us the paragraph that came before this, we won't be able to work out what this is all about. What are x, y, and θ?

5. Jul 23, 2012

### Integral

Staff Emeritus
Re: physics textbook, replace sine with its definition (???)

What is the fundamental definition of sin?

6. Jul 23, 2012

### 1MileCrash

Re: physics textbook, replace sine with its definition (???)

Ill post the rest when I'm at a computer. Thanks again.

7. Jul 23, 2012

### Whovian

Re: physics textbook, replace sine with its definition (???)

Well, there are multiple ways to define it. The ones I can think of off the top of my head are:

$$\sin'=\cos$$
$$\cos'=-\sin$$
$$\sin\left(0\right)=0$$

Another is the unit circle definition, another is:

$$\sin\left(x\right)=\sum_{i=0}^\infty\left(\dfrac{x^{2\cdot i+1}\cdot\left(-1\right)^i}{\left(2\cdot i+1\right)!}\right)$$
$$\cos\left(x\right)=\sum_{i=0}^\infty\left(\dfrac{x^{2\cdot i}\cdot\left(-1\right)^i}{\left(2\cdot i\right)!}\right)$$

(Might have gotten one of those wrong)

And then

$$\arcsin\left(x\right)=\int_0^x\left(\dfrac{\mathrm{d}x}{\sqrt{1-x^2}}\right)$$
$$\arccos\left(x\right)=\int_x^1\left(\dfrac{\mathrm{d}x}{\sqrt{1-x^2}}\right)$$
(Might have gotten that second one wrong, but I'm pretty confident about it)

8. Jul 23, 2012

### nonequilibrium

Re: physics textbook, replace sine with its definition (???)

The formula that is being referred to is $\theta = \arctan \frac{y}{x}$. Other forms are $\sin(\theta) = \frac{y}{\sqrt{x^2+ y^2}}$ and $\cos(\theta) = \frac{x}{\sqrt{x^2+ y^2}}$, which can be derived from the previous. These last two formulas are the definitions the text is referring to.

If you don't understand these formulas and what they're all about, google goniometric circle.

9. Jul 24, 2012

### jbriggs444

Re: physics textbook, replace sine with its definition (???)

What Integral is aiming for is not a derivation for sine that would come from differential calculus or one that would come from a summing of series. He's talking about geometry.

If you have a right triangle, what is the relationship between one of the acute angles and the lengths of the sides?

10. Jul 24, 2012

### Whovian

Re: physics textbook, replace sine with its definition (???)

Ah. The geometric definition. (And those aren't derivations at all, they can be used as definitions.)

11. Jul 25, 2012

### Integral

Staff Emeritus
Re: physics textbook, replace sine with its definition (???)

In that whole long post you never once mentioned the most basic and fundamental definition.

Don't lose track of the basics.