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)])
G += point((x0,y0),size=80,color='red')
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) ) = $'+' ( %s'%str(x0.n(digits=4))+', %s'%str(y0.n(digits=4))+' )'))
if do_f:
G.show()
H.show()
else:
G.show()
Click to the left again to hide and once more to show the dynamic interactive window |
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s,t = var('s,t')
@interact
def _(rad=slider(0.3,1.3,0.01,1,label='$\\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()
Click to the left again to hide and once more to show the dynamic interactive window |
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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)
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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 $y = sin(t)$')
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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)
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%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 $x = cos(t)$')
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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)
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