20 Sept 2010

God is the cause of all essences. Spinoza, Ethics, Book 1, Proposition 25, with Deleuze's Commentary

[The first three subsections are my own overview notes. Under the Spinoza Ethics heading, there again is blue text, which is quotation from the English translation. My summary and commentary to this is bracketed in red. Deleuze's commentary is found near the end. The Latin text comes last.]

What's this proposition got to do with you?

What makes something what it is- this we call its 'essence'. We ourselves have an essence too. We might wonder what causes an essence to be what it is. Why are we some person and not someone else? Perhaps we might dream about being a different person, having his or her essence instead of our own.

Now examine some object near you. We might consider the causes that brought it into existence. This thing also expresses its essence, what it is. Other things can be thought of as causing the existence of this thing. But what causes its essence? Could it be that the fundamental stuff of reality is infinitely expressive, and this thing is one particular way that the underlying substance is expressing itself? As an expression, the thing and its cause are not removed from one another. The cause of the thing's essence would be right there, as its fundamental reality. And is it not possible to think of what underlies of all of what is real in this world-- can it not be thought-of as God? In a way, everything in the world around us expresses God, or at least we can say, anything in our world expresses what underlies all real things in the universe.

Now, we too are things in the world. Also perhaps we might regret our own limitations. We might wish that the essence of who we are would be somehow greater. But let's not forget, our essence has a cause, a reason for being what it is. This cause is infinite and eternal, and yet immediate to us. At every moment, no matter how weakened our essence becomes, we are expressing the infinite entirety of all reality.

Brief Summary:

God is the cause of all essences.

Points Relative to Deleuze
[Under Ongoing Revision]

Things in this world exist only because difference creates the logical 'room' for their being. What allows something to be what it is, is its internal differentiations from itself. If something were self-same throughout itself, in a completely homogeneous way, then there would be nothing there but undefined form. Consider something simple like a circle. It would seem to be very homogeneous. But each point on the edge can only be a part of the circle by being different from the other points. In a sense, what makes a circle is the sum aggregate of all its internal self-differings, all the differential relations between its parts. So we might think that all reality is built on difference, and all real things express difference as their immediate ground. Difference is the immediate cause of anything's self-differing essence.

Baruch Spinoza
Part I "Concerning God"
Proposition XXV

PROP. XXV. God is the efficient cause not only of the existence of things, but also of their essence.

[God is the reason for the existence of each thing. He is also the reason for the essence expressed by every existing thing.]

Proof .— If this be denied, then God is not the cause of the essence of things ; and therefore the essence of things can (by Ax. iv.) be conceived without God. This (by Prop. xv.) is absurd. Therefore, God is the cause of the essence of things. Q.E.D.

[Let's consider the opposite argument to see if it leads to absurdities. So let's assume that god is not the cause of things' essences. Recall that knowing an effect means knowing its cause. So if God is not the cause of things, then we could know the things without also conceiving of God.

Yet, we already established that there is only one substance, God. Other than substance, there are only modes. Modes are conceived through the substances they modify. With there only being one substance, God, all modes then must be conceived through God. Hence it cannot be that modes are thought without also thinking God. So it cannot be that God is not the cause of the essence of things, for if really he were not, then we would not be able to think him while thinking of things' essences. Thus God is the cause of all thing's essences.]

Note .— This proposition follows more clearly from Prop. xvi. For it is evident thereby that, given the divine nature, the essence of things must be inferred from it, no less than their existence-in a word, God must be called the cause of all things, in the same sense as he is called the cause of himself. This will be made still clearer by the following corollary.

[God is infinite. He expresses infinite essence. There could then be no thing's essence which is not inferrable from his essence. It is on account of God's infinite essence that there may be any thing's essence. He then is the cause of all essences.]

LinkCorollary .— Individual things are nothing but modifications of the attributes of God, or modes by which the attributes of God are expressed in a fixed and definite manner. The proof appears from Prop. xv. and Def. v.

[As we noted above, all things exist in God and are conceived through Him. Because they are conceived through Him, and not through themselves, they would then be modifications of God's attributes.]

From Deleuze's Commentary:

Modes are, in their turn, expressive: “Whatever exists expresses the nature or essence of God in a certain and determinate way” (that is, in a certain mode). [ft.3: E I.36p [439] (and 25c: Modi quibus Dei attributa certo et determinato modo exprimuntur).] [Deleuze, 13-14; 352d]

Le mode, à son tour, est expressif : « Tout ce qui existe exprime la nature de Dieu, autrement dit son essence, d’une façon certaine et déterminée » (c’est-à-dire sous un mode défini). [ft.3 : E, I, 36, dem. (et 25, cor. : Modi quibus Dei attributa certo et determinato modo exprimuntur)] [Deleuze, 10-11; 11d]

God produces as he exists; necessarily existing, he produces. [ft.3: E I.25s: “God must be called the cause of all things in the same sense in which he is called the cause of himself”[431]. ] [Deleuze, 100b; 364a]

Dieu produit comme il existe ; existant nécessarement, il produit nécessairement [ft.3 : E, I, 25, sc. : « Au sens où Dieu est dit cause de soi, il doit être dit aussi cause de toutes choses. »] [Deleuze, 88b ; d]

According to Spinoza, God is cause of himself in no other sense than that in which he is cause of all things. Rather is he cause of all things in the same sense as cause of himself. [ft.20: E I.25s. It is odd that Lachièze-Rey, when citing this passage, inverts the order, as though Spinoza had said that God was cause of himself in the sense that he was cause of things. Such a transformation of the passage is not just an oversight, but amounts to the survival of an “analogical” perspective that begins with efficient causality (cf. Les Origines cartésiennes, pp.33-34] [Deleuze, 164a; 375a]

Selon Spinoza, Dieu n’est pas cause de soi en un autre sens que cause de toutes choses. Au contraire, il est cause de toutes choses au même sens que cause de soi [ft.20 : E, I, 25, sc. Il est curieux que P Lachièze-Rey, citant ce texte de Spinoza, en inverse l’ordre. Il fait comme si Spinoza avait dit que Dieu était cause de soi au sens où il était cause des choses. Dans la citation ainsi déformée, il n’y a pas un simple lapsus, mais la survivance d’une perspective « analogique », invoquant d’abord la causabilité efficiente. (CF. Les Origines cartésiennes du Dieu de Spinoza, pp.33-34).] [Deleuze, 149a ;d]

Modal essences therefore, no less than existing modes, have efficient causes. “God is the efficient cause, not only of the existence of things, but also of their essence.” [ft.7, citation] Deleuze, 193c]

C’est pourquoi les essences de modes n’ont pas moins une cause efficiente que les modes existants. « Dieu n’est pas seulement cause efficiente de l’existence des choses, mais encore de leur essence. » [ft.7, citation] [Deleuze, 175]

[from ft.8, Ch.12] The whole of I.24 appears, then, to me to be organized thus: 1. The essence of a thing produced is not cause of the thing’s existence (Proof); 2. But nor is it cause of its own existence as essence (Corollary); 3. Whence I.25, God is cause even of the essences of things. [Deleuze, 378-379]

[from ft.8, Ch.12] L’ensemble de I, 24, nous paraît donc s’organiser ainsi : 1°) l’essence d’une chose produite n’est pas cause de l’existence de la chose (démonstration) ; 2°) mais elle n’est pas davantage cause de sa propre existence en tant qu’essence (corollaire) ; 3°) d’où I, 25, Dieu est cause, même de l’essence des choses. [Deleuze, 176d]

An attribute expresses itself in three ways: in its absolute nature (its immediate infinite mode), as modified (its infinite mediate mode) and in a certain and determinate way (a finite existing mode). [ft.1: E I.21-25] [Deleuze, 235b; 385c]

Un attribut s’exprime de trios façons : il s’exprime dans sa nature absolue (mode infinie immédiat), il s’exprime en tant que modifié (mode infini médiat), il s’exprime d’une manière certaine et déterminée (mode infinie existant). [ft.1 : E, I, 21-25] [Deleuze, 214b ; d]

From the Latin text:


Deus non tantum est causa efficiens rerum existentiæ, sed etiam essentiæ.


Si negas, ergo rerum essentiæ Deus non est causa; adeoque (per Axiom. 4) potest rerum essentia sine Deo concipi: atqui hoc (per Prop. 15) est absurdum. Ergo rerum etiam essentiæ Deus est causa. Q.E.D.


Hæc Propositio clarius sequitur ex Propositione 16. Ex ea enim sequitur, quod ex data natura divina, tam rerum essentia, quam existentia debeat necessario concludi; &, ut verbo dicam, eo sensu, quo Deus dicitur causa sui, etiam omnium rerum causa dicendus est, quod adhuc clarius ex sequenti Corollario constabit.


Res particulares nihil sunt, nisi Dei attributorum affectiones, sive modi, quibus Dei attributa certo, & determinato modo exprimuntur. Demonstratio patet ex Propositione 15, & Definitione 5.


Deleuze, Gilles. Spinoza et le problème de l'expression. Paris: Les Éditions de Minuit, 1968.

Deleuze, Gilles. Expressionism in Philosophy: Spinoza. Trans. Martin Joughin. New York: Zone Books, 1990.

Spinoza. Ethics. Transl. Elwes. available online at:
A fantastic hyperlinked version, thanks Terry Neff:

Spinoza. Ethica. available online at:

The DeComposition of Our Body's Functions. Review and philosophical application of Compound Functions in Edwards & Penney's Calculus (Section 1.4)

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

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[I your author am not a mathematician; I am merely an admirer of Edwards & Penney's wonderful calculus book. Please consult the text or other references to be certain about anything in the summary below. I mean this emphatically.]

[Other Entries in the Edwards & Penney's Calculus Series]

The DeComposition of Our Body's Functions

Review and philosophical application of
Edwards & Penney's
Compound Functions
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

What Do Compound Functions Got to Do With You?

Consider if we are trying to lose weight. So we go on a diet. Over the course of a day, we begin to feel our appetite increase. So this would be one function: as time increases, our appetite increases. But then, as our appetite increases, there is a decrease in our willpower to hold strictly to the diet. So our decrease in willpower is like a function compounded on the first function. Our lives are filled with many such highly complex functional relations.

Brief Summary:

Functions can be compounded one-upon-another as if the output of one machine instantly becomes the input of another.

Points Relative to Deleuze
[Under Ongoing Revision]

We might find a composition of functions in Deleuze's account of sensation, building from his portrayal of Spinoza's affection. Deleuze's example is that we are looking for our glasses in a dark room. Someone enters and turns on the light. This causes us to make a transition from the way we are affected by the dark to the way we are affected by the light. Because we are searching for our glasses, this affection increases our power to act. Consider instead if we are meditating in the dark room. Then when someone turns on the light, this dazzles us and ruins our concentration, and we lose the power to act.

Affection for Spinoza is a matter of our
body's composition. What gives our body its power is the differential relations between the speeds of its simple bodies. But its simple bodies are like calculus differentials, infinitely small parts that only obtain a finite value when put into a differential relation with other simple bodies. Consider if we take poison. This will disrupt the ways the simple bodies in our body differentially relate to one another. This is why we die. Our bodies no longer express the range of differential variations that define us. Instead, they eventually express for example dirt.

Now, Deleuze also places a value on our internal decomposition when it comes to the sensations that affect us. The discord of our faculties is what makes sensations more intense [see here and here.] Also consider Spinoza's Ethics Part IV, Proposition 39, Scholium. He writes:
It sometimes happens, that a man undergoes such changes, that I should hardly call him the same. As I have heard tell of a certain Spanish poet, who had been seized with sickness, and though he recovered therefrom yet remained so oblivious of his past life, that he would not believe the plays and tragedies he had written to be his own : indeed, he might have been taken for a grown-up child, if he had also forgotten his native tongue. If this instance seems incredible, what shall we say of infants? A man of ripe age deems their nature so unlike his own, that he can only be persuaded that he too has been an infant by the analogy of other men. (Elwes translation)
From what Spinoza says about the decomposition of the body, it would seem that to change from child to man, or from poet to someone different, would be a decomposition, and thus a loss of power. But could it be that Deleuze takes the opposite view? To change from child to man is not an expression of our body's frailty. It rather shows our body's power to become something new. Imagine we get a sickness like the Spanish poet, and we no longer recognize our past selves. This would seem to be a loss of power. But that would only be so when we take the perspective of the self whom our body no longer expresses. But our bodies are always as perfect as they can be, given the conditions affecting them at any given moment. So to become someone else is not really a deprivation. It could be a testament to the power of our bodies to become new things, to express new essences. The question for Deleuze might not so much be "What can a body do?" as much as it might be "What can a body become?"

So here would be the composition of functions in this case. A certain affection might for Deleuze increase the intensity of sensation. But that means a decrease in the organization of our faculties and our body's compositional relations. Yet this again would be an increase in our body's power to become new things, to express new relations and essences. So it ultimately is an increase in our body's expressive power.

Edwards & Penney's Calculus
Compound Functions
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

Recall the function machine.

We can think of functions like these 'black box' machines. The machine takes-in a number from one set of values, and it gives-out a value from another set. So consider the squaring function:

The machine takes-in a 2, for example, and then gives-out a 4.

We can also compound our functions. So we can take the result of one function, and then put it into another 'function machine', if you will.

So let's begin with the function g(x). We put in x, and it gives-out u. We then have another function f(u), which is the same as f(g(x)), and we read it as, 'f of g of x'. So we plug the u into the f function, and it gives out the value for f(u). We can then call f(u), or f(g(x)) a new name. We can call it h(x). So,

h(x) = f(g(x))

This requires that all the x values in g's domain correspond (by means of g) to values in f's domain. We call the g function here the 'inner function', and the f function would then be the 'outer function' in this case.

Edwards & Penney now give an example that will illustrate something about notation for composition of functions.

We will first find f(g(x)). First note the g(x). If our number for x is larger than 1, then we will have a negative number. But when we apply it to the f function, we will have the square root of a negative number. So instead we qualify it:

But let's instead consider if we first take f(x), and then after that apply it to the g function. To avoid having the square root of a negative number, we must first begin with a positive. Then after that, there are no restrictions for the g function. So

Edwards and Penney would like then to distinguish two notations:



f g

Our example above shows that the composition of functions f and g is not the same as g and f.

f g g f

So the notation f(g(x)) might remind us of multiplication notation. In the case of multiplication, it does not matter what order the functions are multiplied. But as we see in the composition of functions, the order does matter sometimes.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp. 36-37.

Spinoza. Ethics. Transl. Elwes. available online at:

19 Sept 2010

The Cycles of Our Life-Sines. Trigonometric Functions in Edwards & Penney's Calculus (Section 1.4)

presentation of Edwards & Penney's work,
Corry Shores

Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Mathematics, Calculus, Geometry, Entry Directory]
[Calculus Entry Directory]
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[I am not a mathematician; I am merely an admirer of Edwards & Penney's wonderful calculus book. Please consult the text or other references to be certain about anything in the summary below. I mean this emphatically.

The material here is from Edwards & Penney. Whenever it is not, I will indicate the source. Thanks:
Edwards & Penney
Math Demos

The Cycles of Our Life-Sines
Sine wave circle animation
Review and philosophical application of
Edwards & Penney's

Trigonometric Functions
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

What Do Trigonometric Functions Got to Do with You?

Our lives course-by to many wave-like periodic patterns. Each day the sun rises-and-falls, increasing-and-decreasing light like a wave, with noon as the peak, and the middle of the night is the wave's bottom trough. And those of us living away from the equator experience another sort of waving of the light. Days get longer as we near summer solstice, and shorter as we head toward winter solstice. This can affect our moods, and as well the ways we feel about ourselves, the way we behave, the thoughts we have, and the way we relate and interact with others, loved-ones especially. On winter solstice, the darkness might feel heavy, like we are trapped under a blanket of black. Here we are at the bottom arc of the periodic curve. It might feel as if we are stuck forever in the winter darkness. Now consider more closely that exact moment of winter solstice when the earth enters its magic place along its revolution around the sun. We come upon the limit of the darkening trend. For a moment-out-of time, we no longer tend toward shorter days. And it is not yet until the next moment that the trend will change us toward longer days. Then as we near spring equinox, we can sense the tendency-toward-light increase. We can actually feel these tendencies immediately, as we witness the earth come alive again, erupting from its frozen slumber. And perhaps we too feel forces pushing our moods to the lighter and freer feelings of the summer.

There is something quite remarkable about these wave-like periodic patterns that we find in so many ways in our everyday experiences. They can be described by trigonometry and circles. At any moment in the cycle, a circle is being implicitly expressed. In a way, repetition is there from the beginning, and it is there at every instant. The wave patterns in the world and in our lives express circles in motion. There is an eternal recurrence in our lives, but it is not just the cycles repeating over-and-over seemingly without the intention of ending. Rather, recurrence is already there from the beginning, because the whole circle is implied at the start and it is implied in any instant. In a sense, the eternal form of the circle, and especially its internal structural relationships between its parts, is like a mythical eternal ground for our world and for our lives. It is eternal not because it happens infinitely, but because it is outside of time. All particular recurrences that we see in cyclical patterns happening through time were made possible on the grounds of an eternal recurrence that is outside of time and yet is implicitly immediate to every real moment of its expression in our lives.

Now, the circle just by itself is an expression of eternal repetition. The circle as it begins its motion implies already its return to its beginning. So consider the wave-like patterns which express a circle. The circle they express is eternal, because at any point along the waving change, the circle as a closed shape is already implied. In other words, the circle has returned upon itself even as its wave-expression begins. But this repetition is not possible on the basis of some sameness. The eternal is not a line that goes on forever. It is continually bending away from where it is, but back toward where it began. But when it returns back to where it began, it does not stop there. When it returns, the tendencies toward change are just as strong then as they were at every other place in the circle. And at every point along its changing course, the path is differing from itself already. The change is continuous because the motion always differs from itself. In other words, it is only because self-difference is always in the circle that there is the potential for it to return to itself. So we might think that the circle is everywhere the same. But we see that it is also everywhere tending away from itself, even while tending back toward itself in the same motion, like the snake biting its tail. In our own lives, we go from summer to summer. And it seems like each summer is similar to the rest. But what we experience is not a continual summer. What we experience is a continual self-differing as the changes proceed. We keep having summers only because they never stay the same. It would be one eternal summer if the circle's repetition were based on sameness. Rather, it is a matter of difference already there immediate to us that allows summer to tend away from itself, and welcome itself once again immediately upon itself. We only experience something again if we experience a different instance. We might find ourselves acting out a habitual action that makes us relate to ourselves in the past. But we can only encounter our past selves by meeting ourselves as other to ourselves. The identification is secondary, a matter of recognition. What is primary is that we are firstly different from ourselves.

So consider if we go from experience a to experience b; we can afterward say they are different. But we could not have arrived at b unless first difference paved the way. And we could not have begun at a unless a's way was already paved by difference. Difference is what is most immediate. The things that come to be called different are a secondary derived form of difference. But when we note that we go from summer to summer, from a to a, we should realize again that the second summer, the second a, first needed difference before it could arise. The same for the circle. We cannot say that it arrives back upon its beginning unless the beginning is already different from itself. A circle can begin and end at the same point, but only if that same point is different from itself, so it can serve on the one hand as the beginning, and on the other hand as the end. Now consider how the geometrical form of the circle implies this motion, but itself is not rotating. Any point along the circle is already a beginning and end, a self-different point, a point that treats of itself as being different from itself. Now, what the cycles of our life and world imply is a geometrical circle. Such circles might express themselves in durational wave-like patterns, but as a formal element to that pattern, the circles are outside duration; they are eternal. The circle then should hardly evoke in us the idea of eternal sameness. It is eternal difference, an always and everywhere self-difference.

So as we notice cycles in our lives, we could on the one hand think about how things are somehow both same and different, how there are always ambiguities. We can in addition consider that when we for example lie down to sleep, we are not just sleeping again, or having a different sleep, but that we are immediately experiencing an eternal repetition of difference. Sure, each sleep is unique. But it expresses a difference more profound than its uniqueness. Each sleep is an expression of the difference which made it possible. The eternal substance of all our experience, of everything in our world, is difference.

So if there is one thing that is always repeated eternally, it is difference. The eternal circles implied in the many waves in the world and in our lives are testiments to a more profound eternity, the eternal difference, the eternal return of difference.

Brief Summary

The relations between the sides and angles of a right triangle can be described through the science of trigonometry. We may then apply these relations to angles in circular variation. By doing so, we are able to describe the wave-like periodic patterns in the natural world.

Points Relative to Deleuze
[Under Ongoing Revision]

For Deleuze, our sensations are matters of continuously varying waves. Affection in general is a matter of such a continuous variation. We wonder, why a wave? Is there not continuity to a wave, and thus not complete difference and heterogeneity? Or is there something heterogeneous and discontinuous in the series of tendency-changes along a wave? We should try to be specific about why Deleuze speaks of these waves of sensation, especially considering that sensation is a matter of difference and a discontinuity in our experiences. I currently suspect that the wave is a useful image, because on the one hand there is a continuity in the series; each point or limit is immediately near its neighbor. However, on the other hand there is a discontinuity in the series of tendencies (differentials), moving from one limit to the next along the wave's curves.

Edwards & Penney

Appendix C: Review of Trigonometry

We begin looking at a right triangle.

Edwards Penney Right Triangle adjacent opposite hypoteneuse

We see the right angle in the bottom right corner. The other two must then total 90 degrees, so they both must be acute (less then 90 degrees). The trigonometric functions will be ratios between sides of angles, relative to the acute angle in question. We may arbitrarily pick which of the two acute angles to be θ (theta). Then, the side opposite to theta will be considered the "opposite" side. There are two lines that will come out of theta. One will be opposite to the right angle. This is always the "hypotenuse". The other line extending alongside theta is called the "adjacent" side. We now make ratios of the lengths of the sides. For example, we might take the ratio of the opposite angle over the hypotenuse. By doing so, we have defined the "sine" of theta. When we instead take the ratio of the adjacent line over the hypotenuse, we have then determined the "cosine" of theta. We will use the following abbreviations:
cos = cosign
sin = sine
tan = tangent
sec = secant
csc = cosecant
cot = cotangent
There are six primary trigonometric functions:

edwards penney trigonometric functions secent cosecent tangent cotangent sine cosine

Here by the way is a table calculating the values for our approximations [click to enlarge]

We might also consider these angles more dynamically. They could be formed by a rotation of sorts.

edwards penney circle sine

An angle created by rotation is called a directed angle. Using the image above, we imagine that first the two lines extending from the angle both begin together along the right side of the x axis. Then, one of the lines rotates away while the other remains on the x axis. If the rotation of the line moves counterclockwise, we consider the angle to be a positive angle. But if it goes clockwise, we instead consider it to be a negative angle. We consider P(x,y) to be the point where the moving (terminal) side of theta intersects with the circle. This circle we define as

Let's break from Edwards & Penney to give more background on the circle. Its formula is:

The center-point is (a, b), and r is the radius. To make things easier, we will just make the center-point (0, 0). This way we simplify the formula to.

circle formula

To form a 'unit circle', we make the radius one unit, that is, equal to 1.

unit circle formula diagram

We can construct the circle from this formula. See how the points plotted above fulfill it.

unit circle formula diagram

Now let's pick some point (x, y) on the circle.

unit circle formula diagram

We could then drop a line down from this point to the x-axis.

unit circle formula diagram

It will have the value y, because it extends directly up the y axis. Then we can get the x value by extending the line from that new corner to the origin.

unit circle formula diagram

Then finally the radius line going out to our point on the circle will be 1, because we have defined the radius that way.

unit circle formula diagram

Using the Pythagorean theorem, we see that we return again to our circle formula:

unit circle formula

So let's consider angle theta.

unit circle formula diagram

And recall the formulations for cosine and sine.

formula cosine sine

We see that they are both set over the hypotenuse. But the hypotenuse is 1. The adjacent angle is x. And the opposite angle is y. Hence we can define the following [and we return now to Edwards & Penney]:

We make some assumptions in the above. For tan θ and sec θ, we assume that x ≠ 0 [so that we do not have a number divided by zero]. And for cot θ and csc θ, we assume that y ≠ 0.

If we make substitutions in the above equations, we can use sine and cosine to define the others.

Edwards & Penney then have us compare the angles θ and -θ in this figure:

Notice that the x and y lines will be the same length. Hence:

cos(–θ) = cosθ


sin(–θ) = –sinθ

Then, by working on the values, they give us the following formulas.

The Fundamental Identity of Trigonometry:

cos2θ + sin2θ = 1


1+ tan2θ = sec2θ

thus also

1 + cot2θ = csc2θ

The Addition Formulas:

sin(α + β) = sin α cos β + cos α sin β

cos(α + β) = cos α cos β – sin α sin

The Double-Angle Formulas:

sin 2θ = 2sinθ cosθ

cos 2θ = cos2θ – sin2θ

= 2cos2θ – 1

= 1 – 2sin2θ

The Half-Angle Formulas
(These, they say, are particularly important in integral calculus):

cos2θ = ½(1 + cos2θ)

sin2θ = ½(1 – cos2θ)

Radian Measure

Normally we measure measure angles in degrees, like 360°, 90°, etc. However, there is another means called radian measure. It is often more convenient, and even at times essential, to use radian measure in calculus. So consider this image below [first one from wikipedia]

arc length radius radian
[Thanks wikipedia.org]

arc radius radian circle edwards penney

Here we see that the radius has a certain length, and it equals the circle's arc created by the angle. We say that the angle 'subtends' in the arc. And the length of the arc equals the length of the radius. This wonderful animation from zonalandeducation.com shows how the lengths are equal.

Now consider this animation. It shows that when the diameter of circle is 1 unit, the circumference is pi (π) units.

circumference diameter pi radian animation rolling circle wheel
[Thanks wikipedia.org]

So the diameter multiplied times π will tell us the circumference. This means also that the formula for the circumference could be rendered, when in terms of the radius:

C = 2πr

So the radius measure, times , will give us the circumference. But we are also dividing that circumference up into radius length arcs. Each one corresponds to a fixed angle measure, which we can convert into degrees if we wish. So we know that the circumference is times the radius measure. That corresponds to the total of all the internal angle-measure, which is 360°. If times the radius gives us 360°, then π times radius gives us 180°.

On this basis, we can calculate conversions between radians and degrees.

Here are some conversion charts [The second one is from Edwards & Penney]:

degree radian conversion chart
[Thanks sparknotes.com]

degree radian conversion chart

degree radian conversion circle diagram
[Thanks blog.ssis.edu.vn]

degree radian conversion circle diagram
[Thanks wikimedia.org]

Recall also the formula for the area of a circle:

formula area circle

From this we can obtain the following formulas

Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

Trigonometric Functions

Now first consider this 30-60-90 degree triangle.
30-60-90 degree triangle
[Thanks wikimedia.org]

We see that the sine for the 30 degree angle will be a/2a, or 1/2. And using our conversion methods, we see that this is equal to π/6 radians.

So we see for example these equivalences charted on tables such as these:

trigonometry trigonometric conversion chart degree radian sine cosine tangent
[Thanks cliffsnotes.com]

Now consider this diagram:

Here it is in radian measures:

unit circle to sine wave conversion diagram
[Thanks mathimage.com]

See how the 30 degrees (π/6) on the sine of the circle corresponds to the 0.5, or half-way up the sine wave to the right? A great animation for how the sine wave is formed from the circle values can be seen here.

I tried to capture it, but the quality is much better at the source. [Also it is a wonderful source for many other acoustic related things ideas].

Here is another great animation from Math Demos [Also much better seen at its source]:

And this one is from wikipedia.

circle to sine wave animation
[Thanks wikipedia.org]

And here is one for cosine:

For tangent,

circle to tangent wave animation
[Thanks wikipedia.org]

And for cosecant:

circle to cosecant wave animation
[Thanks wikipedia.org]

Let's return to Edwards & Penney. Using our radians to degree conversions, we obtain:

Now let's also consider when we let the sine wave continue. It repeats its regular oscillation

This is called the periodicity of the trigonometric functions. Every the unit circle rotation returns to where it began. So too does the sine wave pattern repeat its period with each return. Hence

sin(x + 2π) = sin x and cos(x + 2π) = cos x

When we shift ('translate') the cosine graph by π/2, we get the sine graph.

Example 2:
Consider this graph.

It charts the average daily temperatures in Athens, Georgia, USA. It begins in the middle of July, and returns to the next middle of July. As we would expect, the temperatures are higher in the summer, and lower in the winter. They swing like a wave. In this case, the pattern resembles a cosine wave. The formulation that describes this wave is [where average temperature is T in °F, and the number of months after July 15 are t]:

So let's consider October 15, which is three months after July 14. We will then substitute 3 for t. What we obtain is 61.3°F. Note first that the substitution gives us cos(3π/6 ). This is the same as π/2. Recall from the chart that this cos (π/2) = 0.

So now working out the equation is easier:

And of course, this pattern is periodic, which means that in 12 months it returns to where it began.

Let's look again at the graph for y = tan x.

graph for y = tan x

[Thanks wikipedia.org]

The tangent is the blue segment's length divided by the green segment's length. When we near the π/2 part, we have a number divided by an increasingly small number. So 1/ 0.000001 is 1,000,000. As the smaller number nears zero, the quotient of the two nears infinity, hence the tangent lines flying-off to infinite heights. This also means, however, that unlike the sine and cosine graphs, the tangent graph has 'infinite' gaps. Where the top part leaves-off is infinitely high, and where the bottom part picks-up immediately after is infinitely low. Such gaps are called discontinuities. We will later discuss them in more depth.

from Edwards & Penney: Calculus
. New Jersey: Prentice Hall, 2002, pp. A13-15; 33-36.

Image Credits:
Nearly all from Edwards & Penney, except,

Table of Natural Trigonometric Functions:

Triangle in the circle:

Large radian:

Radian animation:

Diameter wheel:

Radian table:

1st radian conversion circle:

2nd radian conversion circle:

30-60-90 degree triangle:

Radian trigonometry conversion chart 1:

Radian trigonometry conversion chart 2:

Radian trigonometry conversion chart 3:

Sine and Cosine with Circle:

Unit circle to sine wave:

3D sine wave animation:

Math Demos animations all from:

Wiki animations from: