Understanding Sine, Cosine and Tangent Angles

[pamichel]

Hi people,

what is meant by sine, cosine, tangent angles, apart from their general definition of opp side/ hypo side; adj side/ hyp side and so on?

how are these sine, cosines and tangent angles invented by humans?

 

[HallsofIvy]

There are a variety of ways to define sine,cosine, etc other than the trigonometric definition (opp side/hypotenuse, etc.) but that is how they were "invented".

They are not angles, by the way, they are functions of angles. Many centuries ago, far too many to be certain exactly how it happened, people noticed that as long as the angles of triangles were the same, while the sides could be as long or short as you wanted, the ratios stayed the same. It happens that right triangles (one angle is always a right angle) were simplest so sine, cosine, tangent were defined as those constant ratios for different angles.

 

[da_willem]

I guess they were 'invented' to use them in the general definition you state. Giving the relation between angles and sides of a triangle.

They also have the intesting property that when you differentiate them twice they are (with a minus sign) the same function, i.e. the slope of a cosine is minus a sine and the slope of a sine is a cosine:

$$\frac{d}{dt} cos(t) =-sin(t)$$ and $$\frac{d}{dt} sin(t) =cos(t)$$

This accounts for the fact that they often occur as a solution of a differntial equation. E.g the equation

$$\frac{d^2 x(t)}{dt^2}=-kx(t)$$

has a linear combination of simes and cosines as a solution because of this property. This equation describes a harmonic oscillator which occur a lot in nature (a spring e.g.). This accounts for the fact that a lot of things move (approximately) sinusoidal.

There is this relation you can derive

$$sin(t)^2=\frac{1}{2}-\frac{1}{2}cos(2t)$$

which means that when you square a sine this yields a cosine shifted along the vertical axis a little and with a doubled frequency. As you might know, the cosine and the sine are of exactly the same shape only shifted a little along the horizontal axis. This means that squaring a sine (or a cosine) doen't fundamentally changes it's shape. This is also a quite extraordinary property.

 

[SGT]

Greeks didn't use sines and cosines, but they used the quotient of the chord and the radius of the circle. They had tables of chords in the same way we used tables of sines and cosines before the advent of electronic calculators.
In a unit circle, the chord of an angle is twice the sine of half the angle.

 

[mathwonk]

i think of them as inverse functions of arclength. i.e. look at a unit circle, and consider the function A such that A(y) = the length of arc on the unit circle measured from (1,0) to the point with second coordinate y.

then sin is the inverse of this function.

the greeks did look at arclength, but perhaps not as a function of length along the y or x axis, although they did consider it for the value y = 1, i.e. A(1) = pi/2.

 

[robert Ihnot]

I think it helps to start with the use of these things. The Greek Thales, who was a rich trader, got the idea of Geometry from the Egyptians who had to survey the land after the Nile flooded its banks. (The Egyptians were pretty good at keeping records for such things as ownership and taxes.)

Now, if you wanted to know how wide a river was, or how high a tree or mountain, it might have been impractical to directly measure it, but you could determine angles and look at similar triangles. Also its good to know about the 3,4,5 triangle and a right angle. But, to go further with it, it is necessary to introduce the notion of proof. After all, how can you be sure that a 3,4,5 triangle has an actual right angle? That is where the Greeks came in.

 

[robphy]

Another way to think about the sine function is that it is the imaginary-component [the odd-component] of $$e^{i\theta}$$. Cosine is the real [and even] component.

 

[fliptomato]

Complex Numbers and Trig Functions--why?

Ah--okay. So we can define $$\sin$$ and $$\cos$$ as ratios of a right triangle, or equivalently cartesian coordinates of a point on a unit circle. This has a nice geometric interpretation. However, at some point, we also prefer to define these trigonometric functions in a more general way to accommodate complex numbers:

$$\sin(x) = \frac{1}{2}(e^{ix}-e^{-ix})$$
$$\cos(x) = \frac{1}{2}(e^{ix}+e^{-ix})$$

One of the advantages of this formalism (aside from giving a natural definition for the hyperbolic trig functions) is that we can recover Euler's formula trivially:

$$\cos(x) + i \sin(x) = e^{ix}$$

However, now I'm curious how we can relate this to the previous geometric interpretation. Instinctively, perhaps, we could recite what is commonly taught and "interpret" the imaginary exponential $$e^{i\theta}$$ as a point on the Cartesian plane with length one from the origin and angle $$\theta$$ from the positive $$x$$-axis.

However, this is just an awkward definition--why do we interpret this complex exponential this way? I have an algebraic understanding of the exponential of real numbers.

$$e^3= e\cdot e\cdot e$$

And by the laws for multiplication of exponentials, we can figure out how to appropriately define the exponential of a negative number, of zero, and even of rational and irrational numbers. However, this is all based on an understanding of the real numbers as things that are closely related to counting numbers, and in a sense we're counting how many times the base of the exponential is being multiplied to itself.

How then, can we understand the exponential of an imaginary number as a polar coordinate, therefore allowing us to connect our generalized definition of the trigonometric functions to our geometric intuition about them?

Perhaps put in another way, why is it that robphy's comment is true? (that the sine is the imaginary component of a complex exponential while the cosine is the real component). We can set this true by definition, but then how do we connect this to a geometric interpretation? (then you say that we can graph the complex exponential as polar coordinates--to which i ask why is a complex exponential supposed to be interpreted as polar coordinates?)

-Flip

 

[yourdadonapogostick]

here is my famous trig thread from scienceforums.net. http://www.scienceforums.net/forums/showthread.php?t=9608 it is my notes from the trig class i took last year. for some reason law of sines doesn't seem to show up. here it is: $$\frac{A}{sin\alpha}=\frac{B}{sin\beta}=\frac{c}{sin\gamma}$$

 

[robphy]

Concerning the exponential approach, recall that one can expand the exponential function in a Taylor series:
$$\displaystyle e^{(i\theta)}=1+(i\theta)+\frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} +\cdots$$
which (it can be shown) can be rearranged into the form $$C(\theta)+iS(\theta)$$ where C and S are real functions. By inspection or by using the complex-conjugate operation, one can verify the parity of these functions. Using the algebraic properties of exponents, one can obtain various angle-sum formulas. By taking derivatives, one can obtain the usual relations between these functions and their derivatives.

Related to the law of sines is
the definition of sine via the cross-product
$$\sin\theta_{between}=\displaystyle\frac{\vec A \times \vec B}{|\vec A| |\vec B|}\cdot\hat z$$
where $$\hat z$$ is a unit-vector perpendicular to the plane spanned by nonzero vectors A and B.
Cosine is analogously defined by
$$\cos\theta_{between}=\displaystyle\frac{\vec A \cdot \vec B}{|\vec A| |\vec B|}$$

 

[SGT]

Ah--okay. So we can define $$\sin$$ and $$\cos$$ as ratios of a right triangle, or equivalently cartesian coordinates of a point on a unit circle. This has a nice geometric interpretation. However, at some point, we also prefer to define these trigonometric functions in a more general way to accommodate complex numbers:

$$\sin(x) = \frac{1}{2}(e^{ix}-e^{-ix})$$
$$\cos(x) = \frac{1}{2}(e^{ix}+e^{-ix})$$

One of the advantages of this formalism (aside from giving a natural definition for the hyperbolic trig functions) is that we can recover Euler's formula trivially:

$$\cos(x) + i \sin(x) = e^{ix}$$

However, now I'm curious how we can relate this to the previous geometric interpretation. Instinctively, perhaps, we could recite what is commonly taught and "interpret" the imaginary exponential $$e^{i\theta}$$ as a point on the Cartesian plane with length one from the origin and angle $$\theta$$ from the positive $$x$$-axis.

However, this is just an awkward definition--why do we interpret this complex exponential this way? I have an algebraic understanding of the exponential of real numbers.

$$e^3= e\cdot e\cdot e$$

And by the laws for multiplication of exponentials, we can figure out how to appropriately define the exponential of a negative number, of zero, and even of rational and irrational numbers. However, this is all based on an understanding of the real numbers as things that are closely related to counting numbers, and in a sense we're counting how many times the base of the exponential is being multiplied to itself.

How then, can we understand the exponential of an imaginary number as a polar coordinate, therefore allowing us to connect our generalized definition of the trigonometric functions to our geometric intuition about them?

Perhaps put in another way, why is it that robphy's comment is true? (that the sine is the imaginary component of a complex exponential while the cosine is the real component). We can set this true by definition, but then how do we connect this to a geometric interpretation? (then you say that we can graph the complex exponential as polar coordinates--to which i ask why is a complex exponential supposed to be interpreted as polar coordinates?)

-Flip

In case you haven't noticed, the two series $$C(\theta)$$ and $$C(\theta)$$ in robphy's post are:
$$C(\theta) = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} + - ... = cos(\theta)$$

$$S(\theta) = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} + - ... = sin(\theta)$$
from there you derive Euler's formula.
Not only $$e^{i\theta}$$ but any complex number can be represented in cartesian or in polar form. In cartesian coordinates:
z = x + iy
in polar coordinates we can define a modulus ( the distance from z to the origin) and an argument (the angle the segment from the origin to z forms with the x axis).
$$|z| = \sqrt{x^2 + y^2}$$

$$arg(z) = tan^{-1}\frac{y}{x}$$

for $$z = e^{i\theta}$$ we have

$$|e^{i\theta} = \sqrt{cos^2\theta + sin^2\theta} = 1$$

$$arg(e^{i\theta} = tan^{-1}\frac{sin\theta}{cos\theta} = \theta$$

 

[Cheman]

My question has always been why is the tangent function called thwe tangent function?

 

[robphy]

My question has always been why is the tangent function called thwe tangent function?

Great question.

wikipedia says http://en.wikipedia.org/wiki/Tangent (and here's a picture in support of its Trigonometric definition http://aleph0.clarku.edu/~djoyce/java/trig/tangents.html)
That may be its historical formulation. (Any historians of mathematics care to comment?)

In the above, the angle theta determines what length is intercepted on the tangent line at (1,0) on a unit circle.


However, I [prefer to] think of it this way:
Given a tangent line to a point on a curve,
the slope of that tangent line = tangent( angle of that tangent line with the horizontal axis).

Is this interpretation in accord with the spirit of the first interpretation?
In other words, if the first is the historical formulation, were they thinking of the second when defining "tangent"? Or is the second somehow a fortunate coincidence in terminology? (References?)

 

[mathwonk]

i recommend reading my post again and thinking about it. it is much more elementary than the other posts here.

 

[robphy]

i recommend reading my post again and thinking about it. it is much more elementary than the other posts here.

Agreed.
Certainly, it's less complex :tongue: .

It's also very natural because it uses the definition of the angle as
(arc length of an arc on the circle)/(radius of that circle).