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.
