1. MEASUREMENT OF ROTATION

2. OBJECTIVES Given an angle of any measure, draw a picture of that angle. Sketch an angle that has a measure of between 0° and 360° and calculate the measure of the reference angle. Find the coterminal angle

3. KEY WORDS & CONCEPTS Initial position Terminal position Standard position Counterclockwise clockwise Coterminal angles (Φ=θ+ 360n°) Reference angle Degrees, minutes, seconds

4. DEFINING MEASUREMENT OF ROTATION In the Ferris wheel problem of Section 2-1, you saw that you can use an angle to measure an amount of rotation. In this section you will extend the concept of an angle to angles whose measures are greater than 180° and to angles whose measures are negative. You will learn why functions such as your height above the ground are periodic functions of the angle through which the Ferris wheel turns.

5. DEFINING MEASUREMENT OF ROTATION An angle as a measure of rotation can be as large as you like. For instance, a figure skater might spin through an angle of thousands of degrees. To put this idea into mathematical terms, consider a ray with a fixed starting point. Let the ray rotate through a certain number of degrees, θ, and come to rest in a terminal (or final) position. Terminal position Initial position Rotatin g ray Fixed point Angle θ So that the terminal position is uniquely determined by the angle measure, a standard position is defined. The initial position of the rotating ray is along the positive horizontal axis in a coordinate system, with its starting at the origin. Counterclockwise rotation to the terminal position is measured in positive degrees, and clockwise is measured in negative degrees.

6. WARM UP A function that repeats its values at regular intervals is called a _______________ function. Describe the transformations g(x) = 5 f(x) g(x) = 4 + f(x) g(x) = 4 + f(x) 40 is 20% of what number? periodic y-dilation by a factor of 5 y-translation by + 4 200

7. DEFINITION STANDARD POSITION OF AN ANGLE An angle is in standard position in a Cartesian coordinate system if: -Its vertex is at the origin -Its initial side is along the positive horizontal axis -It is measured counterclockwise from the horizontal axis if the angle measure is positive and clockwise from the horizontal axis if the angle measure is negative.

8. EXAMPLES Several angles in standard position are shown in the graphs below. A uv-coordinate system (v for vertical) is used for plotting angles, θ, as measures of rotation, with both u and v depending on θ. The customary variable x and y will be used for plotting sinusoids, with the angle appearing as a displacement along the horizontal axis in the graphs.

9. EXAMPLES

10. EXAMPLES The same position can have several corresponding angle measures. For instance 493° angle terminates in the same position as the 133° angle after one full revolution (360°) more. The -227 angle terminates there as well, by rotating clockwise instead of counterclockwise. The graphs show these coterminal angles.

11. GREEK LETTERS Letters such as θ may be used for the measure of an angle or for the angle itself. Other Greek letters are often used as well: a (alpha), β (beta), (gamma) ϕ (phi) (pronounced “fye” or “fee”), ϖ (omega). You might recognize some of the Greek letters on this subway sign in Athens, Greece.

12. EXAMPLE 1 Sketch angles of 71°, 133°, 254°, and 317° in standard position and calculate the measure of each reference angle. Solution: to calculate the measure of the reference angle, sketch an angle in the appropriate quadrant; then look at the geometry to find out what to do. The graphs show the four angles along with their reference angles. For an angle between 0° and 90° (in Quadrant 1), the angle and the reference angle are the same. For angles in other quadrants, you have to calculate the positive angle between the x-axis and the terminal side of the angle. Note that if the angle is not between 0° and 360°, you can first find a coterminal angle that is between these values.

13. EXAMPLE 2 Sketch an angle of 4897° in standard position, and calculate the measure of the reference angle. Solution: The number tells you that the terminal side makes 13 whole revolutions plus another 0.6027…revolution. To find out which quadrant the angle falls in, multiply the decimal part of the revolutions by 360 to find the number of degrees. The answer is a coterminal angle to θ between 0 ° and 360 °. Divide 4897 by 360 to find the number of whole revolutions. Compute without rounding.

14. SOLUTION CONTINUED Sketch the 217° angle in Quadrant III as below. From the graph you should be able to see that As you draw the reference angle, remember that it is always between the terminal side and the horizontal axis (never the vertical axis). The reference angle sometimes goes from the axis to the terminal side and sometimes form the terminal side to the axis. To figure out which way it goes, recall that the reference angle is positive. Thus it always goes in the counterclockwise direction.

15. CH. 2.2 Assignments Textbook pg 65 #2, 6, 10, 12, 16, 26, 28 & 30