The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is thought of as a purely mechanical process (e.g. "make sure your calculator is in degrees and put this in your calculator when you want to find out the length of this side of a triangle from this other side of a right triangle".)

My first question to you is:

  • How do you introduce the basic trigonometric functions to students?

As a corollary to this question, I ask:

  • What is your definition of the basic trigonometric functions?

Outside of this usage to calculate (1) the side lengths of triangles, students also learn early on to define these functions as (2) models for periodic motion; and later on as (3) coordinates on the unit circle and (4) the analytic sum of some power series (or some other limiting approximation of the classical power series definition of sine and cosine).

With these four definitions in mind, I ask these additional questions.

  1. What methods are used to discuss the differences and similarities between these four (and possibly more) definitions of trigonometric functions?

  2. What methods are there for motivating the connection between a geometric definition of the trigonometric functions and an analytic definition of the trigonometric function?

  3. How are these definitions (1) and (4) connected in a (late high school or early college) student's calculus course?

  4. Is it useful for students to see the connection between definitions (1) and (4), or is it enough to provide them with the tools and knowhow to use them and just let them use them as they are commanded to in their science classes?

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    $\begingroup$ That could be a great question if it where more detailed. What levels are you considering? Which context? Are you interested in having a rigorous treatment without logical loops (which is rarely what students are exposed to, since trig functions are spread over many years)? As it is, it probably deserves closing, but improvements could change that. $\endgroup$ – Benoît Kloeckner Mar 5 '16 at 17:08
  • $\begingroup$ The details from when I originally asked the question have been reintroduced. Does it satisfy the requirements needed to take it off hold? Is it the goal of this website to foster dialogue on those questions and answers which reveal our constant need to demand clarity and exactness in our thinking, or is this a place where authority (whether used wisely or not) consumes argument? $\endgroup$ – John Mar 9 '16 at 18:52
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    $\begingroup$ it does for me even if part of it are a bit unfocused to my taste; but then it is a question that makes sense and can attract interesting answers . At least five different people did not thought your original question had these qualities, and you could have considered this as valuable information rather than an occasion to rant against authority. $\endgroup$ – Benoît Kloeckner Mar 9 '16 at 19:28
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    $\begingroup$ I have cleaned up the organization and questioning. I hope I have captured the essence of your question still. I took some liberties where there were gaps in the initial description. To the original author @John: Please let me know if you feel uncomfortable with these changes. I think in this current form it should be taken off of hold. $\endgroup$ – Andrew Sanfratello Mar 10 '16 at 5:32
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    $\begingroup$ @John An additional comment for your comment above. This site is, as defined in the tour "all about getting answers. It is not a discussion forum. There's no chit chat." Obviously, chit chat happens, but I think the reason your question was put on hold was for it's lack of clarity $\endgroup$ – Andrew Sanfratello Mar 10 '16 at 5:38

How do you introduce the basic Trigonometric functions to students?

In my geometry courses, as in most American secondary school geometry courses of which I am aware, the sine and cosine functions are introduced as ratios of lengths of sides of right triangles.

The tangent function can be introduced simply as a ratio of those two ratios, however the tangent function is realizable as a geometric length if one considers a similar right triangle to that from which the angle comes such that the hypotenuse is unity. The value of tangent of the angle is the signed length of the vertical line segment that is tangent to the unit circle and whose endpoints lie on the x-axis and the terminal side of the angle.

Given a right triangle, label measures of the acute angles $\alpha$ and $\beta$, the length of the side opposite $\beta$ label $b$, and length of the side opposite $\alpha$ label $a$. Label the length of the hypotenuse $h$. Then $$ \sin \alpha = \frac{a}{h} $$ $$ \cos \alpha = \frac{b}{h}$$ $$ \sin \beta = \frac{b}{h} $$ $$ \cos \beta = \frac{a}{h}$$

Notice that cosine of an acute angle gives the "complementary sine," or the sine of the complementary angle. Because every right triangle is similar to a triangle with hypotenuse one (multiply the length of each side by $\dfrac{1}{h}$ ), and because corresponding angles of similar triangles are congruent, we can discuss the sine or cosine of an angle without reference to any particular right triangle.

First in geometry, students learn to compute the sine or cosine of an acute angle in a right triangle by writing a ratio of the triangles sides.

In precalculus algebra, after students have been familiarized with functions, the definition of these ratios is extended to include right and obtuse angles, by describing angles in the unit circle. By relating any angle to acute or right angles in the first quadrant via reference angles, the student can then find the vertical or horizontal displacement of an infinite number of arbitrarily close points on the unit circle without the use of a computational device after establishing identities.

What is your definition of the basic Trigonometric functions?

Formally, trigonometric functions take equivalence classes of real numbers as their argument, with the equivalence class representative given by the real number modulo $2\pi$.

Consider a ray initiating at the origin. Since the ray is continuous and has length greater than 1, the ray must intersect the unit circle at some point, say $(x,y)$. Define the measure of the angle to be the length of the arc of the circle that initiates at $(1,0)$ and terminates at $(x,y)$. A degenerate angle has a measure of 0, and every other angle has a measure in the interval $(0,2 \pi)$, since the circumference of the unit circle is $2 \pi$. Using these definitions, $\sin(\theta)$ maps the interval $[0,2\pi)$ to the value of $y$ associated with the intersection of the ray and the unit circle. $\cos(\theta)$ maps the interval $[0,2\pi)$ to the value of $x$ associated with the intersection of the ray and the unit circle.

Once these functions have been defined as functions of an angle, one can consider arguments less than $0$ or greater than $2 \pi$. For real numbers $\alpha$ and $\beta$, let $\alpha$ relate to $\beta$ if $\alpha \equiv \beta \mod 2 \pi$. This is an equivalence relation and thus gives equivalence classes over the real numbers, each with a unique representative in $[0,2 \pi)$. Then for $\theta < 0$ or $\theta \geq 2 \pi$, let $\sin\theta = \sin (\theta \mod 2 \pi)$. A similar definition is given for cosine.

After the functions have been extended over the real numbers, it is useful to consider the case that the angle parameter $\theta$ is a linear function of time. Such functions are immediately applied to describing the dynamic of a simple harmonic oscillator. By considering alternative parameterizations of the angle parameter, one can obtain useful function models of a number of oscillating systems that yield important results in many areas of science.

Furthermore, once analytic definitions are demonstrated, as detailed below, there is no difficulty associated with considering a complex parameter, whose value is given as a complex number in terms of the hyperbolic sine and hyperbolic cosine functions. So, it is even reasonable to discuss complex angles, which is convenient, for example, when considering elements of a vector space over a field of complex numbers.

What methods are their for motivating the connection between the geometric definition of the trig functions and the analytic definition of the trigonometric function?

With my students, I relate these functions to power series through Euler's identity $$e^{i \theta} = \cos \theta + i \sin \theta.$$

I introduce this identity in precalculus without explicit proof. By that time, my students have discovered complex numbers as zeros of polynomials. Initially in precalculus, we discuss the roots of unity in rectangular form $a + ib$ without any mention of the polar form $e^{iθ}$ or the trig functions. When we introduce the trig functions and unit circle later in the course, I include a discussion of the polar form $e^{iθ}$ and show that the polar numbers also solve those polynomials, and we see that the two forms are equal.

Euler's identity, when considered in the context of the complex unit circle, has incredible pedagogical value as it constitutes a unifying theme of algebra and geometric trigonometry.

How are these two definitions connected in a student's calculus course?

The series definition for the exponential function is demonstrated early in my calculus course. Returning to the long established relationship between the complex exponential function and the trigonometric functions, one can obtain the series representations of $\sin \theta$ and $\cos \theta$ by begining with the series representation for $e^x,$ with which most students are comfortable, then substituting $x = i \theta$ in that series. After some elementary algebraic simplification, one obtains a sum of two series, one a real series of even power functions, and the other an imaginary series of odd power functions. Returning to the original statement of the identity, the students then discover that the even series must be the power series representation of the cosine function, and the odd series must be the representation of the sine function.

Is it useful for students to see a connection between these two different definitions, or is it enough to give them the tools and let them use them as they are commanded to in their science classes?

For students who wish to study science, mathematics, and engineering, an understanding of the geometric and analytic properties of the trigonometric functions and how they emerge from study of the complex unit circle are indeed crucial. Students will encounter many methods for modeling and problem solving in later courses and in their industries that presume such an understanding. Some examples are the Laplace Transform integral solution of a differential equation, Fourier Analysis, electric currents and fields, vectors and vector analysis, representations of cyclic groups, matrix representations of linear transformations, and the list goes on.

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    $\begingroup$ Isn't Euler's identity usually obtained from the Taylor series of the exponential, sine and cosine functions, as opposed to the other way around? $\endgroup$ – J W Mar 5 '16 at 6:55
  • $\begingroup$ @JW It doesn't need to be. I introduce Euler's identity before calculus, so by the time my students see Taylor series they are already very familiar with complex numbers in polar form and Euler's identity. $\endgroup$ – Andrew Mar 8 '16 at 20:24
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    $\begingroup$ How do you prove Euler's identity? (See also math.stackexchange.com/questions/3510/…) $\endgroup$ – J W Mar 8 '16 at 21:01
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    $\begingroup$ Personally, though, I would start with showing them DeMoivre's formula by having them graph $(1+i)^n$ from $n=0$ to $n=10$ and make a table of $\theta$ values and $|(1+i)^n|$ values. $\endgroup$ – Simply Beautiful Art Mar 12 '16 at 0:13
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    $\begingroup$ @JW If you have introduced $e^x \approx (1+\frac{x}{n})^n$ for large values of $n$, then it should make sense that $e^{ix}\approx (1+\frac{ix}{n})^n$. $1+ix \approx \cos(\frac{x}{n})+i\sin(\frac{x}{n})$ geometrically. Then DeMoivre takes you home. $\endgroup$ – Steven Gubkin Mar 12 '16 at 14:52
  1. I was introduced by the unit circle. This seems simpler than the ratio of sides since hypotenuse is 1, although the ratio of sides is more general and physically meaningful. But it is just simpler to grok initially. To this day, I mentally picture the unit circle and picture a radar scope with the line moving counterclockwise. Might be "bad" but I did fine in trig and further classes. [It is probably an easier picture also when thinking about "negative angles" or angles greater than 180 or 360. See the "sweep" returning in your mind.]

  2. I think having physics class soon after trig class is VERY helpful since trajectories and such make such great use of trig in terms of vectors. It gives good practice, motivation, and mental picture of the angles.


I'm really surprised by the way that some people here start defining trigonometric functions: as all those functions can have positive as well as negative values, it makes (in my humble opinion) no sense to define them as result of the division of two positive numbers. The way I learnt them started with the basis on the unity circle, in which an angle was drawn, and from that, the basic functions were defined (sin(x) as a the coordinate on the Y-axis and cos(x) as the coordinate on the X-axis).
I must admit that one thing was teached to me in the complete wrong way: the definition of the tangent of an angle (it was teached as being the division of sin(x) and cos(x)). I only realised how wrong this was, once I got faced with a student I was teaching: I had told him that sin(x) is the coordinate on the Y-axis, cos(x) is the coordinate on the X-axis, and tangent(x) was the division of both, to which he answered

"Sir, I don't believe you! If you say that both sin(x) and cos(x) are coordinates, then also tangent(x) must be a coordinate, or I don't believe what you say!"

Although the student was known for his bad behaviour, I must admit he was right: tangent(x) is the coordinate of the angle, as cut off using the tangent line going through the point (1,0).
I do understand that it might be useful knowing the link between trigonometric functions and lengths of sizes of triangles, but this might not replace the pure definitions of the trigonometric functions.


Since it has not been mentioned by name in the other answers, I'll say that once we model the x and y coordinates of a point on the unit circle as $\left(\cos(\theta),\sin(\theta)\right)$ (marked by a ray through the origin), I define the tangent as the slope of that ray. Since they've found the slope of a line so many times before, my students end up drawing a pretty reasonable graph of the function $y=\tan(\theta)$ before knowing the $\frac{\text{opposite}}{\text{adjacent}}$ definition. [But they end up figuring out $\frac{\text{sine}}{\text{cosine}}$ based on their slope definition: $\frac{\text{rise}}{\text{run}}$

Of course, this works because the text we use does everything with the unit circle${}^*$ for a chapter before covering triangles.

[* And I model the circle by pulling the front wheel off my bicycle and putting a sticky-note on the edge of the tire. Spinning the wheel and holding it so it faces the students and is either vertical or horizontal is my go-to for seeing the up-and-down or back-and-forth motion of the sticky-note point.]


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