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 If the radius = 1 and the center of the circle is at 0,0 Then 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 0,0 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 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:

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: hyp

Finding the X Offset hyp Knowing the radius and the angle above or below the horizontal The x offset is found by:

Example #1 Find the x and y offset for point A A

Example #2 Find the x and y offset for point A A

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 A (2,4)

Example #4 For the circle center at 2,4 find the location of point A A (1.325,2.750)

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

Example #5 Enter the desired radius Press tab Enter the desired angle Press enter

Assignment Complete Polar Coordinate wks. #1