# 102 - Topic 03 - Radian Measure with sine and cosine graphs

## 51 days ago by Professor102

John Travis

Mississippi College

Radian measure relates the measure of an angle in standard position to the length of the arc created by that angle on a unit circle.

For a given angle $\theta$ in radians, since the radius for the unit circle is $1$, then

$\sin(\theta)=y$

and

$\cos(\theta)=x$.

In the following interactive experiment, one can approximate the values of $\sin(\theta)$ and $\cos(\theta)$ for any given angle $\theta$ by finding the point of intersection with the unit circle.

Below, enter the desired angle in radians.  Use the resulting coordinates (shown in the experiment) for the sine and cosine.

Plotting $x = \sin(s)$ and $y = \cos(s)$ is done by seting $\alpha = s$.  The graphic illustrated the relationship between this radian measure angle $\alpha$ with the arclength $s$

%hide %auto # trig functions using endpoint of unit circle arc # also, creating the graph y=sin(x) using definition # angle in radians s,t = var('s,t') @interact(layout=dict(top=[['rad'],['Graph']])) def _(rad=slider(-10,10,1/10,1,label='$$\\alpha =$$'), Graph=['none','sine','cosine'],auto_update=True): x0 = cos(rad) y0 = sin(rad) G = circle((0,0),1,color='blue',alpha=0.2) G += line([(0,0),(x0,y0)]) do_f = True if Graph=='sine': f = sin elif Graph=='cosine': f = cos else: do_f = False if do_f: H = plot(f(s),(s,-10,10),color='blue',alpha=0.2,ymin=-1,ymax=1) # Now, draw the "angle" as an increasing spiral if rad>0: G += parametric_plot(((t/60+1/10)*cos(t), (t/60+1/10)*sin(t)), (t, 0, rad), color='blue') G += parametric_plot((cos(t),sin(t)),(t,0,rad),color='red',thickness=5) if do_f: H += plot(f(s),(t,0,rad),color='red',thickness=5) else: G += parametric_plot(((abs(t)/60+1/10)*cos(t), (abs(t)/60+1/10)*sin(t)), (t, rad, 0), color='blue') G += parametric_plot((cos(t),sin(t)),(t,rad,0),color='red',thickness=5) if do_f: H += plot(f(s),(s,rad,0),color='red',thickness=5) pretty_print(html('$( \cos(\\theta),\sin(\\theta) ) =$'+' ( %s'%str(x0.n(digits=4))+', %s'%str(y0.n(digits=4))+' )')) if do_f: G.show() H.show() else: G.show()

$$\alpha =$$
Graph

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

s,t = var('s,t') @interact def _(rad=slider(0.3,1.3,0.01,1,label='$\\alpha =$'),grid = checkbox(default=False)): x0 = cos(rad) y0 = sin(rad) t0 = tan(rad) cot0 = 1/t0 G = point( (x0,y0), color='red',size=30,zorder=3)+circle((0,0),1,color='blue',alpha=0.2,ymin=0,xmin=0) G += line([(x0,0),(x0,y0)],thickness=2,color='black')+line([(0,0),(0,y0)],thickness=2,color='black') G += text('sin',(-0.04,y0/2),rotation=90)+text('sin',(x0+0.04,y0/2),rotation=90) G += line([(0,y0),(x0,y0)],thickness=2,color='black')+line([(0,0),(x0,0)],thickness=2,color='black') G += text('cos',(x0/2,y0+0.04))+text('cos',(x0/2,-0.04)) if t0>cot0: G += line([(0,0),(1,t0)],thickness=6,color='lightgray',zorder=0.5) else: G += line([(0,0),(cot0,1)],thickness=6,color='lightgray',zorder=0.5) G += line([(0,0.01),(1,t0+0.01)],color='blue')+text('sec',(0.96,t0+0.02),color='blue',horizontal_alignment='right') G += line([(0,-0.01),(cot0,0.99)],color='purple')+text('csc',(cot0-0.02,0.96),color='purple',horizontal_alignment='left') G += line([(1,0),(1,t0)],color='green')+text('tan',(1.1,t0/2)) G += line([(0,1),(cot0,1)],color='orange')+text('cot',(cot0/2,1.1)) G += parametric_plot(((abs(t)/60+1/10)*cos(t), (abs(t)/60+1/10)*sin(t)), (t, rad, 0), color='blue') G += parametric_plot((cos(t),sin(t)),(t,rad,0),color='red',thickness=5) gridmarks = [.1,.2,.3,.4,.5,.6,.7,.8,.9,1,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2] n=20 num_grids = ceil(max(t0,cot0)) g = [k/n for k in range(n*num_grids)] if grid: G.show(gridlines=[g,g]) else: G.show()

$\alpha =$
grid

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

Let's now create an animation...

s,t = var('s,t') GraphAnim = [] num = 500 for j in range(num): rad = j/num+0.3 A = "Angle (in radians) = %4.3f"%(rad) x0 = cos(rad) y0 = sin(rad) t0 = tan(rad) cot0 = 1/t0 G = point( (x0,y0), color='red',size=30,zorder=3)+circle((0,0),1,color='blue',alpha=0.2,ymin=0,xmin=0,title=A) G += line([(x0,0),(x0,y0)],thickness=2,color='black')+line([(0,0),(0,y0)],thickness=2,color='black') G += text('sin',(x0+0.04,y0/2),rotation=90) # G += text('sin',(-0.04,y0/2),rotation=90)+text('sin',(x0+0.04,y0/2),rotation=90) G += line([(0,y0),(x0,y0)],thickness=2,color='black')+line([(0,0),(x0,0)],thickness=2,color='black') G += text('cos',(x0/2,y0+0.04)) # G += text('cos',(x0/2,y0+0.04))+text('cos',(x0/2,-0.04)) if t0>cot0: G += line([(0,0),(1,t0)],thickness=6,color='lightgray',zorder=0.5) else: G += line([(0,0),(cot0,1)],thickness=6,color='lightgray',zorder=0.5) G += line([(0,0.01),(1,t0+0.01)],color='blue')+text('sec',(0.96,t0+0.02),color='blue',horizontal_alignment='right') G += line([(0,-0.01),(cot0,0.99)],color='purple')+text('csc',(cot0-0.02,0.96),color='purple',horizontal_alignment='left') G += line([(1,0),(1,t0)],color='green')+text('tan',(1.1,t0/2)) G += line([(0,1),(cot0,1)],color='orange')+text('cot',(cot0/2,1.1)) G += parametric_plot(((abs(t)/60+1/10)*cos(t), (abs(t)/60+1/10)*sin(t)), (t, rad, 0), color='blue') G += parametric_plot((cos(t),sin(t)),(t,rad,0),color='red',thickness=5) GraphAnim.append(G) animate(GraphAnim).show(delay=3000/num, iterations=0)

The next two animations illustrate the relationship between the position of the ordered pair $(x,y)$ on a unit circle with a corresponding point on the graph of the sine and cosine curves respectively.

In particular, the y-coordinate on the unit circle relates to the y-value of the sine function.

$y = \sin(t)$

and the x-coordinate on the unit circle relates to the y-value of the cosine function.

$x = \cos(t)$

The calculation of this animation make take a little while so be patient.  When the green bar to the left below is showing, the calculations are still being performed.

%hide # looping animation creating the sine curve # angle in radians t = var('t') GraphA = [] for r in range(1,126): rad = real(r/10) x0 = -1+cos(rad) y0 = sin(rad) G = circle((-1,0),1.0,color='blue',alpha=0.5)+line([(-1,0),(x0,y0)],figsize=(10,4),axes=false) G += plot(sin(t),(t,0,6.2),color='blue',alpha=0.2,ymin=-1,ymax=1) if rad<2*pi: G += parametric_plot((-1+cos(t),sin(t)),(t,0,rad),color='red',thickness=3) else: G += parametric_plot((-1+cos(t),sin(t)),(t,0,2*pi),color='red',thickness=3) G += parametric_plot((-1+cos(t),sin(t)),(t,2*pi,rad),color='orange',thickness=3) G += plot(sin(t),(t,0,rad),color='blue',thickness=3) G += arrow( (-1+cos(rad),sin(rad)), (rad,sin(rad)) , color='yellow') GraphA.append(G) animate(GraphA).show() html('Graph comparing vertical-coordinate of point $(x,y)$ on unit circle with $y = sin(t)$')
 Graph comparing vertical-coordinate of point (x,y) on unit circle with y = sin(t) Graph comparing vertical-coordinate of point (x,y) on unit circle with y = sin(t)
%hide # looping animation creating the cosine curve # angle in radians t = var('t') GraphA = [] for r in range(1,126): rad = real(r/10) x0 = cos(rad) y0 = 1+sin(rad) G = circle((0,1),1.0,color='blue',alpha=0.5)+line([(0,1),(x0,y0)],xmin=-1,xmax=1,axes=false,figsize=(4,10)) G += parametric_plot((cos(t),-t),(t,0,6.2),color='blue',alpha=0.2) if rad<2*pi: G += parametric_plot((cos(t),1+sin(t)),(t,0,rad),color='red',thickness=3) else: G += parametric_plot((cos(t),1+sin(t)),(t,0,2*pi),color='red',thickness=3) G += parametric_plot((cos(t),1+sin(t)),(t,2*pi,rad),color='orange',thickness=3) G += parametric_plot((cos(t),-t),(t,0,rad),color='blue',thickness=3) G += arrow( (cos(rad),1+sin(rad)), (cos(rad),-rad) , color='yellow') GraphA.append(G) animate(GraphA).show() html('Graph comparing horizontal-coordinate of point $(x,y)$ on unit circle with $x = cos(t)$')
 Graph comparing horizontal-coordinate of point (x,y) on unit circle with x = cos(t) Graph comparing horizontal-coordinate of point (x,y) on unit circle with x = cos(t)