Physics textbook, replace sine with its definition (?)

[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?

 

[Muphrid]

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.

 

[1MileCrash]

Both? Or one or the other?

 

[DrGreg]

"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."

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?

 

[Integral]

What is the fundamental definition of sin?

 

[1MileCrash]

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

 

[Whovian]

What is the fundamental definition of sin?

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)

 

[nonequilibrium]

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.

 

[jbriggs444]

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?

 

[Whovian]

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?

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

 

[Integral]

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

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

Don't lose track of the basics.