locating points on a circle
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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 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 0,0Then point A is at 1,0
A
(0,0)
Finding Points at 90 degrees
If the radius = 1 and the center of the circle is at 0,0Then point B is at 0,1
B
(0,0)
Finding Points at 180 degrees
If the radius = 1 and the center of the circle is at 0,0Then 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 0,0Then point D is at 0,-1 D
(0,0)
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:
hyp
oppsin
hyp
adjcos
adj
opptan
Trig Functions
The functions allow one to find the vertical and horizontal offsets from the center of the circle.
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 horizontalThe y offset is found by:
hypxsinθopp
hypopp
sinθ
radiusxoffsety sin
hyp
θ
Finding the X Offset
hypxcosθa
hypadj
cosθ
dj
radiusxoffsetx cos
hyp
θ
Knowing the radius and the angle above or below the horizontalThe x offset is found by:
Example #1
Find the x and y offset for point A
A590
2.500288.1
500.2)59cos(
cos0
x
radiusxoffsetx
143.2
500.2)59sin(
sin0
x
radiusxoffsety
1.288
2.143
Example #2
Find the x and y offset for point A
596.2
250.3)37cos(
cos0
x
radiusxoffsetx
956.1
250.3)37sin(
sin0
x
radiusxoffsety
A
370
3.250
2.596
1.956
Finding the Point LocationTo 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 #3For the circle center at 2,4 find the location of point A.
590
2.500
288.1offsetx143.2offsety
1.288
2.143
A
(2,4)
),(int ococ yyxxApo )143.24,288.12(
)143.6,288.3(
Example #4For the circle center at 2,4 find the location of point A.
596.2offsetx956.1offsety
),(int ococ yyxxApo )956.1750.2,596.2325.1(
)706.4,271.1(
370
3.250
2.596
1.956
A
(1.325,2.750)
ReviewPolar 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.
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))*B1In the cell below y axis enter =cos(radians(B2))*B1
RADIUS ANGLE X AXIS YAXIS1 0 0 1
Example #5
Enter the desired radiusPress tabEnter the desired anglePress enter
RADIUS ANGLE X AXIS YAXIS3.100 45 2.192 2.192