Locating Points on a Circle

1. Locating Points on a Circle Sine Cosine Tangent

2. Coordinates Systems Review • There are 3 types of coordinate systems which we will use: • Absolute • Incremental • Polar

3. 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.

4. 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.

5. Finding Points • When a line rotates around a point, a circle is created.

6. 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

7. A (0,0) 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

8. B (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

9. C (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

10. (0,0) D 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

11. 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

12. Trig Functions • The functions are a ratio of two of the sides to one of the angles. • The ratios are:

13. Trig Functions • The functions allow one to find the vertical and horizontal offsets from the center of the circle.

14. Trig Functions • The vertical offset = the amount of change on the y axis.

15. Trig Functions • The horizontal offset = the amount of change on the x axis.

16. 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.

17. hyp Finding the Y Offset • Knowing the radius and the angle above or below the horizontal • The y offset is found by:

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

19. A 590 2.500 2.143 1.288 Example #1 • Find the x and y offset for point A

20. A 3.250 370 1.956 2.596 Example #2 Find the x and y offset for point A

21. 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.

22. A 590 2.500 2.143 (2,4) 1.288 Example #3 • For the circle center at 2,4 find the location of point A.

23. A 3.250 370 1.956 (1.325,2.750) 2.596 Example #4 • For the circle center at 2,4 find the location of point A.

24. 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.

25. An Additional Note • This work may also be performed using a spreadsheet.

26. 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

27. Example #5 • Enter the desired radius • Press tab • Enter the desired angle • Press enter

28. Assignment • Complete Polar Coordinate wks. #1