Locating Points on a Circle

SineCosineTangent

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 0,0Then 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

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 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.

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))*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

Assignment

Complete Polar Coordinate wks. #1