Trigonometry: PolarEquations: anIntroduction by: javier
Trigonometry: Polar Equations: an Introduction
by: javier
Polar Equation the difference
the difference
the Cartesian approach to coordinates
the difference
y
(3, 4)
x
the difference
11
12 y
1 5
10
(3, 4) (5, 1 : 05 o′ clock) 2
9
x 4
8 5
7 6
3
the difference
105 120◦ 135◦ 150◦
◦ y90◦
75◦
(3, 4) 60◦
(5, 52o )
45◦ 30◦
165◦
15◦
180◦
0◦ 1 2 3 4 5x 345◦
195◦
330◦ 210◦ ◦ 225 315◦ ◦ ◦ 300 240 255◦ 270◦ 285◦
the difference
y
(x, y) (r, θ) r θ x
y
x
potting points on Polar Equations graph r = 5 cos(θ)
potting points on Polar Equations graph r = 5 cos(θ) θ
r
0 15 30 45 60 75 90 120 150 180
5.0 4.8 4.3 3.5 2.5 1.3 0.0 -2.5 -4.3 -5.0
120◦
◦ 105◦ 90 75◦
135◦
60◦
45◦
150◦
30◦ 15◦
165◦ 180◦
1 2 3 4 5 6 7
195◦
345◦
330◦
210◦ 225◦ 240◦
0◦
315◦
255◦ 270◦ 285◦
300◦
potting points on Polar Equations r = 3 + 5 sin(θ)
potting points on Polar Equations r = 3 + 5 sin(θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
plot by thinking, finding max and min r values r = 5 sin(2θ)
plot by thinking, finding max and min r values r = 5 sin(2θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
plot by thinking, finding max and min r values r = 5 cos(3θ)
plot by thinking, finding max and min r values r = 5 cos(3θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
plot by thinking, finding max and min r values r = 10 sin(5θ)
plot by thinking, finding max and min r values r = 10 sin(5θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
graph by converting to cartesian r = 10 sin(θ)
graph by converting to cartesian r = 10 sin(θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
graph by machine
graph r2 = 4 cos(2x) polar sage
graphing Polar Equations find start and end of a portion of graph
find start and end of a portion of graph r = 7 sin(3θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
find start and end of a portion of graph r = 3 + 7 cos(θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
find start and end of a portion of graph Determine the area that lies inside r = 5 − 3 cos(θ) and outside r = 4 ◦ 120◦
90
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
find start and end of a portion of graph Determine the area that lies inside r = 5 − 3 cos(θ) and outside r = 4 ◦ 120◦
90
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
by: javier
Polar Equation the difference
the difference
the Cartesian approach to coordinates
the difference
y
(3, 4)
x
the difference
11
12 y
1 5
10
(3, 4) (5, 1 : 05 o′ clock) 2
9
x 4
8 5
7 6
3
the difference
105 120◦ 135◦ 150◦
◦ y90◦
75◦
(3, 4) 60◦
(5, 52o )
45◦ 30◦
165◦
15◦
180◦
0◦ 1 2 3 4 5x 345◦
195◦
330◦ 210◦ ◦ 225 315◦ ◦ ◦ 300 240 255◦ 270◦ 285◦
the difference
y
(x, y) (r, θ) r θ x
y
x
potting points on Polar Equations graph r = 5 cos(θ)
potting points on Polar Equations graph r = 5 cos(θ) θ
r
0 15 30 45 60 75 90 120 150 180
5.0 4.8 4.3 3.5 2.5 1.3 0.0 -2.5 -4.3 -5.0
120◦
◦ 105◦ 90 75◦
135◦
60◦
45◦
150◦
30◦ 15◦
165◦ 180◦
1 2 3 4 5 6 7
195◦
345◦
330◦
210◦ 225◦ 240◦
0◦
315◦
255◦ 270◦ 285◦
300◦
potting points on Polar Equations r = 3 + 5 sin(θ)
potting points on Polar Equations r = 3 + 5 sin(θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
plot by thinking, finding max and min r values r = 5 sin(2θ)
plot by thinking, finding max and min r values r = 5 sin(2θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
plot by thinking, finding max and min r values r = 5 cos(3θ)
plot by thinking, finding max and min r values r = 5 cos(3θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
plot by thinking, finding max and min r values r = 10 sin(5θ)
plot by thinking, finding max and min r values r = 10 sin(5θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
graph by converting to cartesian r = 10 sin(θ)
graph by converting to cartesian r = 10 sin(θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
graph by machine
graph r2 = 4 cos(2x) polar sage
graphing Polar Equations find start and end of a portion of graph
find start and end of a portion of graph r = 7 sin(3θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
find start and end of a portion of graph r = 3 + 7 cos(θ) 120◦
90◦
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
find start and end of a portion of graph Determine the area that lies inside r = 5 − 3 cos(θ) and outside r = 4 ◦ 120◦
90
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
find start and end of a portion of graph Determine the area that lies inside r = 5 − 3 cos(θ) and outside r = 4 ◦ 120◦
90
60◦
150◦
30◦
0◦
180◦
330◦
210◦ 240◦
270◦
300◦
