gPRE CALCULUS GRADE 11The Unit Circle.pptx

THE UNIT OF CIRCLE

1. The Unit Circle Grade 11 - STEM

2. Angles and Angle Measurement In geometry, an angle is defined as the union of two non-collinear rays with a common endpoint. The definition restricts the angle measures between 0 and 180 exclusive.  

3. Angles and Angle Measurement In trigonometry, angles will be discussed in a different perspective. Angles will be thought of as any amount of rotation generated when a ray is rotated about its endpoint, hence, the necessity to label the sides of an angle.

4. Angles and Angle Measurement The initial position of the ray will be referred to as initial side of the angle, and the position of the ray at the end of its rotation will be called terminal side of the angle. The arrowhead is pointed at the terminal side. The endpoint of the ray is the vertex of the angle.

5. Angles and Angle Measurement The direction of rotation of a ray also affects angle measure. If the rotation of the ray is counterclockwise, a positive angle is formed. If the rotation of the ray is clockwise, a negative angle is formed.

6. Example: Draw the following angle measure 45 4. 0 ° 150 ° 5. -30° 400° 6. -500°  

7. Angles in Standard Position An angle is said to be in standard position if its vertex is at the point of origin of a Cartesian Coordinate Plane, and its initial side is along the positive side of the x-axis.

8. Quadrantal Angles A quadrantal angle is an angle in standard position whose terminal side lies on the x-axis and y-axis.

9. Coterminal Angles A coterminal angles that are both in standard position and have the same terminal sides.

10. Example: Determine the measure of the smallest positive angle coterminal with the given angle. 42 3. -213 ° 108 ° 4. 635°  

11. Reference Angles The reference angle for a given angle is the positive acute angle formed by the terminal side of the given angle and the x-axis.

12. Example: Find the reference angles of the following: 42 3. 113 ° 221 ° 4. -410°  

13. Units of Angle Measure There are three units of angle measure that are commonly used: Revolution Degree Measure Radian Measure

14. Units of Angle Measure A revolution (rev) is formed when the initial side of an angle rotates completely around its vertex. Consequently, the angle’s terminal side and initial side coincide.

15. Units of Angle Measure The Babylonians developed the degree measure at around 2000 B.C. It was based on a sexagesimal numeration system. To one complete rotation of a ray, or to a revolution, 360 ° is assigned. One degree is equal to 60 minutes, and 1 minute is equal to 60 seconds. 1 rev = 360 ° 1° = 60’ 1’ = 60”

16. Example: Change each angle measure in decimals to minutes and seconds. 1 . 22.4 2 . 14.21 °  

17. Units of Angle Measure One radian (rad) is the measure of a central angle subtended by an arc of a circle that is equal to the radius of the circle.

18. Converting Angle Measure To convert degree measure to radian measure , multiply the number of degrees by , then simplify. To convert radian measure to degree measure , multiply the number of radians by , then simplify.  

19. Example: Convert each radian measure to degree measure. 1 . 2 .  

20. Example: Convert each degree measure to radian measure. 1 . 45 ° 2 . -108 °

21. The Unit Circle A unit circle is a circle whose center is at the origin and with a radius of 1 unit. Every point on the unit circle satisfies the equation . Some of points on the unit circle are ( 1, 0 ), ( 0, 1 ), (-1, 0), and (0, -1)  

22. Example: Determine whether each point is inside, on, or outside the unit circle. 3. 4. (1.1, -0.1)  

23. Thank you for listening