Is the sine function defined at pi/2?

[Orange-Juice]

Wouldn't the sine function be undefined at pi/2 since at that point there would be no triangle in the unit circle, only a straight line along the y-axis? The hypotenuse of a right triangle must always be the longest side of the triangle so I don't see how the sine function can ever give you a value of one. Is it simply taken to be one at pi/2 for the sake of continuity? Thanks

 

[micromass]

You mean $\pi/2$ ?

 

[Orange-Juice]

You mean $\pi/2$ ?

Whoops yeah I mean pi/2

 

[micromass]

I'm sorry, but I really don't follow? Could you make a drawing perhaps.

 

[Orange-Juice]

No I did mean pi/2 you're right, forgot the unit circle was 2pi.

 

[micromass]

OK, so I agree that defining the sine in terms of triangles would not give a good definition then. But consider the following definition:

If $\alpha\in \mathbb{R}$. Let $T$ be the half-line through the origin such that the angle with the $X$-axis is $\alpha$. Let $(x,y)$ be intersection of $T$ with the unit circle. We define $\cos(\alpha) = x$ and $\sin(\alpha) = y$.

In this sense, if $\alpha = \pi/2$, then the half-line in the definition is the positive part of the $Y$-axis. This intersects the unit circle in $(0,1)$. Hence by this definition, $cos(\pi/2) = 0$, $\sin(\pi/2) = 1$.

 

[Orange-Juice]

Ah okay that definition clears things up. The primary (at least in elementary math) use of the trig function is to find ratios of the sides of triangles though right?
So is this definition of the trig functions just used since it makes the functions continuous and differentiable at all points and allows them to be used to model periodic relationships in the sciences and other areas of math?

 

[micromass]

Ah okay that definition clears things up. The primary (at least in elementary math) use of the trig function is to find ratios of the sides of triangles though right?

This would be the only reason why were care about trig functions in elementary math, yes. Of course, if you get more advanced, you will see more and more applications of trig functions. For these other applications, it will become important that the sine functions are extended beyond $(0,\pi/2)$. For example, when looking at vibrations in a string, they will naturally become sine functions.

 

[SteamKing]

No I did mean pi/2 you're right, forgot the unit circle was 2pi.

All circles subtend an angle of 2π, not just the unit circle. It is called a unit circle because the radius = 1 unit.

 

[epenguin]

Then'trigonometric' functions are also called the 'circular' functions - so if you define that on the unit circle sin θ is the height of a point on the circle where the line at angle θ to the horizontal cuts it, and the cosine cos θ is the horizontal distance to the point, then you don't have to think of sin π/2 (or cos 0 ) as being special in this way.