Trig Unit Circle

GeoGebra Unit Circle

The GeoGebra Circle Below is a “Unit Circle”. What makes it so special? The radius is equal to one, so the circumference is 2 п.  An angle has been created that lies in standard position (vertex at origin, one vector on the x-axis oriented to the point (1,0)). The slider creates the potential values for the angle. Angles are measured as we travel counterclockwise from the standard position. Take a minute to play with the slider for Ө (theta). Notice that it can only go between 0 and 360 degrees.

 Can you envision how a negative angle would be represented? Write your on a piece of paper.
 Using this tool, describe on the same piece of paper how could you would calculate a negative angle?
 What positive angle would correspond to 90o ?

You can switch between radians and degrees by selecting options on the top tool bar and then angle unit. We will be working in radians in Calculus, so switch to radian mode. Each radian corresponds to one radius distance around the circumference starting from standard position. There are 2 п radians in a 360 o angle.  П is defined as the ration of circumference to diameter of any circle.

• Can you explain why there are 2 п radians in a 360 o angle?
• Where would п radians be on the unit circle? What positive angle does this correspond to in degrees?
• Trade your paper with another member of the class that you are not sitting with. Read the other person’s explanations and decide if you agree. 
• Where would п/4 radians be? What angle would it correspond to in degrees?

Defining Sine, Cosine, and Tangent with the Unit Circle. 

Sine, and cosine are functions that each take as an argument an angle measure (Ө) and return numeric values. Given an angle in standard position on the unit circle, the cosine will be the x coordinate of the terminating end of the angle, and the sine will be the y coordinate.  Your GeoGebra Unit circle




They can be defined using the unit circle or using a right triangle that contains the angle Ө.

Show 8 Points

45º

90º

135º

180º

225º

270º

315º

0º

Show Degrees

 \frac{ π }{4 } 

 \frac{ 3π }{4 } 

 \frac{ π }{2 } 

 \frac{ 5π }{ 4} 

\frac{ 3π }{ 2} 

 \frac{ 7π }{ 4} 

Show Radians

2π

π

 ( \frac{  \sqrt{ 2 }  }{ 2} ,  \frac{  \sqrt{ 2 }  }{ 2} )

(1, 0)

(0, 1)

 ( - \frac{  \sqrt{ 2 }  }{ 2} ,  \frac{  \sqrt{ 2 }  }{ 2} )

(-1, 0)

 ( - \frac{  \sqrt{ 2 }  }{ 2} ,  - \frac{  \sqrt{ 2 }  }{ 2} )

(0, -1)

 ( \frac{  \sqrt{ 2 }  }{ 2} ,  - \frac{  \sqrt{ 2 }  }{ 2} )

Show Coordinates

Cosine(θ)*

Sine(θ)*

Tangent(θ)*

* values rounded

Show graph

Graph cosine(θ)

Unit circle

Values

Ferris Wheel Unit Circle

( 0 , 0 )

(          ,          )

Show Coordinates (rounded)

0

1

-1

Vertical line

Connect circle & curve

Coordinate point value

Coordinates cos(θ)

Graph sin(θ)

Pivot

Allow pivot

Show Picture

Run/Pause

Pivot

Graph of function

Coordinate point

Hide extra seats

drag

Height
  (m)

100

75

50

25

0

Time

Make graph

Height tool

(clockwise)

(counter-clockwise)

time

Green seat

Graph green seat

makes animation button appear

Speed

D_1 = (-1.02, 0.43)
C_1 = (-1.6, 0)
B_1 = (-2.2, -0.15)
f_1 = CircumcircleArc(B_1, C_1, D_1)
Rotate = Point(f_1)
Center_2: (-1.0 * x(Rotate) - 3.2, 1.3)
Center_1: (x(Rotate) - 1.2, 1.3)
theta = 0.10471975511965978 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
g: 1.0 * x = theta
height = Function(1.2 * sin(x) + 1.3, -0.2, 10.0)
C: (theta, 1.2 * sin(theta) + 1.3)
l_1 = Tangent(C, height)
m_1 = Slope(l_1)
HorizLine: (-1.0, y(C))
j_1 = Line(HorizLine, C)
PtLine_{conic} = (-2.2, 2)
i_1 = OrthogonalLine(PtLine_{conic}, xAxis)
E_1 = Point(i_1)
G_1 = (-2.199, 0.1)
F_1 = (-2.201, 0.1)
k_1 = Conic(E_1, Center_1, F_1, G_1, Center_2)
J_1, L_1 = Intersect(k_1, j_1)
Car = If(x(Rotate) > -1.0 * 2.2, If[m_1 ≥ 0.0, (x(J_1), y(J_1)), (x(L_1), y(L_1))], (1.2 * cos(theta) - 2.2, 1.2 * sin(theta) + 1.3))
height_{line} = Segment(Car + (-1.0 * 0.5, 0.0), C + (100.0, 0.0))
A = (-2.2, 1.3)
b = Segment(A, Center_2)
angle_{pt} = Point(b)
c_{orig-circle} = Circle(A, 1.2)
H_1 = Point(c_{orig-circle})
I_1 = Rotate(H_1, theta, A)
c = Line(A, I_1)
D = Point(c_{orig-circle})
value_{curves}: round((1.0 + theta - 1.0) * 57.25)
s_1: sin(theta)
c_1: cos(theta)
B = Intersect(c_{orig-circle}, xAxis, 2.0)
B' = Rotate(B, theta, A)
θ = Angle(B, A, B')
Y: sin(theta) + 1.0
e: 1.0 * y = sin(theta)
h(x) = x + theta
β = 0 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
D'_1 = Rotate(D, -(β), A)
γ = Angle(D'_1, A, D)
a = Segment(A, Car)
ncos: cos(-(β))
nsin: sin(-(β))
d: 1.0 * x = β
f: 1.0 * y = ncos
E = Intersect(d, f)
i: 1.0 * y = nsin
βsin(x) = -sin(x)
G = Intersect(d, i)
j = OrthogonalLine(Car, b)
H = Intersect(j, b)
I = Point(c_{orig-circle})
k = Segment(Center_2, Center_1)
J = Intersect(k, j)
m = Segment(Car, J)
cosine = Function(1.2 * cos(x) + 1.3, -0.2, 10.0)
F = Intersect(cosine, g, (0.10471975511965978, 2.493426274441928))
K = Point(c_{orig-circle})
L = Point(c_{orig-circle})
M = Point(c_{orig-circle})
N = Point(c_{orig-circle})
O = Intersect(c_{orig-circle}, k, 1.0)
P = Point(c_{orig-circle})
Q = Point(c_{orig-circle})
R = Point(c_{orig-circle})
points = false
degrees = false
radians = false
show_{coordinates} = false
t_1: tan(theta)
show_{cosine} = false
show_{sine} = false
show_{tangent} = false
show_{graph} = false
graph_{cosine} = false
circle = false
values = false
S: (0.01 / 3.0 + theta, 2.62)
T = Point(Circle[S, 2.48])
a_1 = Segment(S, T)
a_2 = Segment(S, T)
U = Intersect(m, k)
t = Segment(U, A)
text31: c_1 + ""
text32: s_1 + ""
text33: c_1 + ""
text34: s_1 + ""
V = Midpoint(U, A)
W = Midpoint(Car, J)
circle_{coordinates} = false
text36: "( " + value_{curves} + " , " + c_1 + " )"
text37: "( " + value_{curves} + " , " + (sin(theta)) + " )"
Z = Point(cosine)
A_1 = Point(cosine)
e_1 = Segment(Car, C)
g_1 = Segment((0.0, 0.1), (0.0, 2.5))
d_1 = Segment((-0.05, 0.1), (0.05, 0.1))
b_1 = Segment((-0.05, 2.5), (0.05, 2.5))
Grid2 = Sequence(Segment[(0.0, 0.6 * i + 0.1), Point[Circle[(6.3, 0.6 * i + 0.1), 0.0]]], i, 0.0, 4.0)
vertical_{line} = false
connect = false
coordinate_{value} = false
coordinates_{cosine} = false
Grid = Sequence(Segment[(0.175 * i, 0.1), Point[Circle[(0.175 * i, 2.5), 0.0]]], i, 0.0, 36.0)
graph_{sine} = false
K_1 = (-0.6351421188630492, 2.2946547931748795)
Car_2 = Rotate(Car, 29.999999999999996*°, A)
Car_3 = Rotate(Car, 59.99999999999999*°, A)
Car_4 = Rotate(Car, 90.0*°, A)
Car_5 = Rotate(Car, 119.99999999999999*°, A)
Car_6 = Rotate(Car, 150.00000000000003*°, A)
Car_7 = Rotate(Car, 180.0*°, A)
Car_8 = Rotate(Car, 210.0*°, A)
Car_9 = Rotate(Car, 239.99999999999997*°, A)
Car_{10} = Rotate(Car, 270.0*°, A)
Car_{11} = Rotate(Car, 300.00000000000006*°, A)
Car_{12} = Rotate(Car, 329.99999999999994*°, A)
pivot = true
n_1 = Segment(A, Car)
p_1 = Segment(A, Car_2)
q_1 = Segment(A, Car_3)
r_1 = Segment(A, Car_4)
b_2 = Segment(A, Car_5)
c_2 = Segment(A, Car_6)
d_2 = Segment(A, Car_7)
e_2 = Segment(A, Car_8)
f_2 = Segment(A, Car_9)
g_2 = Segment(A, Car_{10})
h_2 = Segment(A, Car_{12})
i_2 = Segment(A, Car_{11})
text42: theta * 180.0 / 3.141592653589793° + ""
speed_1 = 1 (slider from -3 to 3, step 0.1)
M_1 = (-3.65995, -0.457)
N_1 = (-0.72837, -0.457)
O_1 = (-3.65995, 2.59)
pix = true
pivot_{backgroundarc} = CircumcircleArc(B_1, C_1, D_1)
α = Angle(angle_{pt}, A, Car)
angle_{pt2} = Point(a)
k_2 = CircleSector(A, angle_{pt}, angle_{pt2})
graph_{function} = false
coordinate_{point} = false
hide = false
make_{graph} = false
text51: 40.0 * (1.25 * sin(theta) + 1.25) + " m"
height_{point} = Intersect(j_2, height_{line})
height_{tool} = false
text52: "theta = " + theta
text56: "( " + (38.2 * x(C)) + " , " + (40.0 * (1.25 * sin(theta) + 1.25)) + " )"
green = false
Car_{green}: (x(Car_{10}), y(Car_{10}))
E_2: (theta, 1.2 * sin(theta - 1.565) + 1.3)
l = Segment(Car_{green}, E_2)
p = Function(1.2 * sin(x - 1.565) + 1.3, -0.2, 10.0)
greenGraph = false
dummyAnimate = -1.9 (slider from -5 to 5, step 0.1)
cosVal: "= " + c_1
sinVal: "= " + s_1
tanVal: "= " + t_1
speed: "= " + (abs(speed_1))
speedCk = false
graphPtH1 = 0 (slider from 0 to 6.3, step 0.1)
graphPtH2 = 0.5234351104782992 (slider from 0 to 6.3, step 0.1)
graphPtH3 = 1.0476958052816296 (slider from 0 to 6.3, step 0.1)
graphPtH4 = 1.57195650008496 (slider from 0 to 6.3, step 0.1)
graphPtH5 = 2.0962171948882906 (slider from 0 to 6.3, step 0.1)
graphPtH6 = 2.63117708754475 (slider from 0 to 6.3, step 0.1)
graphPtH7 = 3.1554377823480806 (slider from 0 to 6.3, step 0.1)
graphPtH8 = 3.679698477151411 (slider from 0 to 6.3, step 0.1)
graphPtH9 = 4.203959171954741 (slider from 0 to 6.3, step 0.1)
graphPtH10 = 4.728219866758072 (slider from 0 to 6.3, step 0.1)
graphPtH11 = 5.252480561561402 (slider from 0 to 6.3, step 0.1)
graphPtH12 = 5.766042058511603 (slider from 0 to 6.3, step 0.1)
graphPtV1 = 0.18822589948131224 (slider from 0.1 to 2.6, step 0.1)
graphPtV2 = 0.18822589948131224 (slider from 0.1 to 2.6, step 0.1)
graphPtV3 = 0.18822589948131224 (slider from 0.1 to 2.6, step 0.1)
graphPtV4 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV5 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV6 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV7 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV8 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV9 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV10 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV11 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPtV12 = 0.19892509733444141 (slider from 0.1 to 2.6, step 0.1)
graphPt1: (graphPtH1, graphPtV1)
graphPt2: (graphPtH2, graphPtV2)
graphPt3: (graphPtH3, graphPtV3)
graphPt4: (graphPtH4, graphPtV4)
graphPt5: (graphPtH5, graphPtV5)
graphPt6: (graphPtH6, graphPtV6)
graphPt7: (graphPtH7, graphPtV7)
graphPt8: (graphPtH8, graphPtV8)
graphPt9: (graphPtH9, graphPtV9)
graphPt10: (graphPtH10, graphPtV10)
graphPt11: (graphPtH11, graphPtV11)
graphPt12: (graphPtH12, graphPtV12)
graphPtH13 = 6.3 (slider from 0 to 6.3, step 0.1)
graphPtV13 = 0.18822589948131224 (slider from 0.1 to 2.6, step 0.1)
graphPt13: (graphPtH13, graphPtV13)