102 - Topic 01 - Angles and Trig Definition

170 days ago by Professor102

John Travis

Mississippi College

Many trigonometry textbooks introduce the trigonometric functions $\sin(\theta)$ and $\cos(\theta)$ using only very basic angle concepts.  Especially when starting with degree measure for angles, the following is how students are introduced to the sine and cosine functions:

          Consider an angle $\theta$ be in standard position and select any point $(x,y)$ on the terminal side. 

          With $r = \sqrt{x^2+y^2} = $ distance along the terminal side from the point to the origin,

$\sin(\theta) = y/r$ and $\cos(\theta) = x/r$

In the following interactive experiment, one can approximate the values of $\sin(\theta)$ and $\cos(\theta)$ for any given angle $\theta$ by finding a "good" point on the terminal side.  Visually, a "good" point may be one that lies on a horizontal (or vertical) grid line as well as on a circle of radius $r$.  In this way, it is easy to measure the horizontal offset $x$ (or vertical offset $y$) and the radial distance $r$.

Below, enter the desired angle--in degrees--and adjust the radius r until the red point on the terminal side lies as close as possible to a x (or y) grid line.  The closer fit the better.  Use the resulting coordinates (shown in the experiment) to approximate the sine and cosine.

You may need to click on the graph to redraw the picture after making any change.

       
$\alpha = $  $r = $ 

Click to the left again to hide and once more to show the dynamic interactive window

Experiment with the interactive cell above and compare your graphically-determined values for various angles vs using an electronic calculator.  Some suggested examples (with really good numerical approximations) are below.

(To make things look good, I have created functions SIN(t), COS(t) and TAN(t) where you can enter t in degrees.  Generally, lower case is required.)

%auto def SIN(t): return sin(t*pi/180).n(digits=4) def COS(t): return cos(t*pi/180).n(digits=4) def TAN(t): return tan(t*pi/180).n(digits=4) 
       
print SIN(143) print COS(165) print TAN(200) 
       
0.6018
-0.9659
0.3640
0.6018
-0.9659
0.3640
%auto # definition of trig functions using point on terminal side only # N: size of grid to display [-N,N] # angle in degrees t = var('t') N = 20 angle = [25+180,69.5+180,111+180,149+180] r = [19,14,17,15] T = ['A','B','C','D'] G = Graphics() for j in range(4): rad = angle[j]*pi/180 Nx = N*cos(rad) Ny = N*sin(rad) x0 = (r[j]*Nx/N) y0 = (r[j]*Ny/N) R = range(1,N+1) G += line([(0,0),(Nx,Ny)]) G += point((x0,y0),size=50,color='red',zorder=1) G += text(T[j],(x0,y0+2),fontsize=15,color='black') for k in R: G += circle((0,0),k,color='lightblue',alpha=0.5) G += line([(k,-N),(k,N)],color='lightgreen',alpha=0.4)+line([(-k,-N),(-k,N)],color='lightgreen',alpha=0.5) G += line([(-N,k),(N,k)],color='lightgreen',alpha=0.4)+line([(-N,-k),(N,-k)],color='lightgreen',alpha=0.4) # Now, draw the "angle" as an increasing spiral if rad>0: G += parametric_plot(((t/6+1)*cos(t), (t/6+1)*sin(t)), (t, 0, rad), color='blue') else: G += parametric_plot(((abs(t)/6+1)*cos(t), (abs(t)/6+1)*sin(t)), (t, rad, 0), color='blue') G.show()