Download Locating Points on a Circle

Locating Points on a Circle
Sine
Cosine
Tangent
Coordinates Systems Review
There are 3 types of coordinate systems
which we will use:



Absolute
Incremental
Polar
Coordinates Systems Review
Absolute

Uses the origin as the reference point for all other
points. Measures location as a distance along the
axis.
Incremental

Uses the present position as the reference point
for the next point. Measures location as a distance
along the axis.
Polar

Use the current location as the reference point.
Measures location as a distance and an angle.
Polar Coordinates
Derives the name from the rotation of a
line around a fixed point.
When this occurs, a circle is formed.
Points may be found on the circle using
the polar coordinate system.
Finding Points
When a line
rotates around a
point, a circle is
created.
Finding Points at 0, 90, 180,
270 degrees
When the line is
at 0, 90, 180 and
270 degrees, the
point may be
found by adding
or subtracting the
radius of the circle
from the center
point of the circle
Finding Points at 0 degrees
If the radius = 1
and the center of
the circle is at 0,0
Then point A is at
1,0
A
(0,0)
Finding Points at 90 degrees
B
If the radius = 1
and the center of
the circle is at 0,0
Then point B is at
0,1
(0,0)
Finding Points at 180 degrees
If the radius = 1
and the center of
the circle is at 0,0
Then point C is at
–1,0
C
(0,0)
Finding Points at 270 degrees
If the radius = 1
and the center of
the circle is at 0,0
Then point D is at
0,-1
(0,0)
D
Trig Functions
Any of the other points located on the
circle may be found using trigonometry.
Trigonometry (trig) is the study of
triangles.
Trig uses 3 functions (equations)



Sine
Cosine
Tangent
Trig Functions
The functions are a ratio of two of the
sides to one of the angles.
The ratios are:
opp
sin 
hyp
adj
cos 
hyp
opp
tan 
adj
Trig Functions
The functions allow one to find the
vertical and horizontal offsets from the
center of the circle.
Trig Functions
The vertical offset = the amount of
change on the y axis.
Trig Functions
The horizontal offset = the amount of
change on the x axis.
Trig Functions
Or if both the x and
y offsets are known,
the angle between
the center of the
circle and the point
on the circle.
Finding the Y Offset
Knowing the radius and the angle above
or below the horizontal
The y offset is found by:
opp
sinθ 
hyp
opp  sinθ x hyp
y offset  sin  x radius
θ
Finding the X Offset
Knowing the radius and the angle above
or below the horizontal
The x offset is found by:
adj
cosθ 
hyp
adj  cosθ x hyp
x offset  cos x radius
θ
Example #1
Find the x and y offset for point A
x offset  cos x radius
 cos( 590 ) x 2.500
 1.288
A
2.500
2.143
y offset  sin  x radius
 sin( 590 ) x 2.500
 2.143
1.288
590
Example #2
Find the x and y offset for point A
x offset  cos x radius
 cos( 37 0 ) x 3.250
 2.596
A
3.250
370
1.956
y offset  sin  x radius
 sin( 37 0 ) x 3.250
 1.956
2.596
Finding the Point Location
To find the point location:


Calculate x and y offset
Add or subtract the values from the circle
center location
 If the point is towards the right of the center,
add the x offset value.
 If the point is towards the left of the center,
subtract the x offset value.
 If the point is above the center, add the y offset
value.
 If the point is below the center, subtract the y
offset value.
Example #3
For the circle center at 2,4 find the
location of point A.
x offset  1.288
y offset  2.143
po int A  ( xc  xo , yc  yo )
 (2  1.288, 4  2.143)
 (3.288, 6.143)
A
2.500
2.143
(2,4)
1.288
590
Example #4
For the circle center at 2,4 find the
location of point A.
A
x offset  2.596
y offset  1.956
3.250
370
1.956
(1.325,2.750)
po int A  ( xc  xo , yc  yo )
 (1.325  2.596, 2.750  1.956)
 (1.271, 4.706)
2.596
Review
Polar coordinates



Uses the current location as the reference
point.
Measures location as a distance and an
angle.
Trig may be used to find the x & y
coordinates of a point given in polar
coordinates.
An Additional Note
This work may also be performed using
a spreadsheet.
Here’s how.
Label 4 cells radius, angle, x axis and y
axis as shown below.
In the cell below x axis enter
=sin(radians(B2))*B1
In the cell below y axis enter
=cos(radians(B2))*B1
RADIUS ANGLE X AXIS YAXIS
1
0
0
1
Example #5
Enter the desired radius
Press tab
Enter the desired angle
Press enter
RADIUS ANGLE X AXIS YAXIS
3.100
45
2.192
2.192
Assignment
Complete Polar Coordinate wks. #1