Learning Goal: Students will understand radian measure of an angle,

1. Learning Goal: • Students will understand radian measure of an angle, • be able to find co-terminal angles of a given angle; • students will be able to determine the reference angle of a given angle. Agenda: Prior Knowledge Check Lesson: explanation slide followed by guided practice problems. Practice angles on the website as directed. Students participate at the interactive whiteboard. Summative Assessment Homework References: Precalculusby Carter, Cuevas, Day, Malloy, Bryan, Holliday, and Hovsepian; Glencoe McGraw Hill Precalculus Graphical, Numerical, Algebraic by Demana, Waits,Foley, and Kennedy; Adison Wesley

2. Prior Knowledge Check Find all the missing sides of each triangle” 1. 2. x 6 1 8 3. 4. 6 A B 5. Which side of the triangle is the largest? 6. Which ratio of sides is the largest? C

3. Why do we Study Trigonometry? When motion of an object causes air molecules to vibrate, we hear sound. Analyzing sound waves is an important application of trigonometry. Trigonometric functions arose from the consideration of ratios within right triangles. In calculus trigonometric functions are very important. Every kind of periodic function (recurring behavior) can be modeled to any degree of accuracy by using trigonometric functions.

4. In geometry, an angle is the union of two rays with a common endpoint. • An angle with its vertex at the origin and it initial side along the positive x-axis is said to be in standard position. • If the terminal side of an angle is rotating in a counter-clockwise direction, the angle formed is a positive angle; • If the terminal side of the angle is rotating in a clockwise direction, the angle is a negative angle. Two angles in standard position, with the same terminal side, are considered coterminal angles. As a terminal side is rotated counterclockwise, its path is a circle. The unit circle is a circle with radius 1.

5. Measuring Angles in Radians and Degrees • The measure of an angle is based on the amount of rotation from the initial side to the terminal side of the angle. • One complete revolution (the terminal side coincides with the initial side) is defined as , so one degree is 1/360 of a complete revolution. • In trigonometry and calculus, a different way of measuring angles, called radians, is used more often than degrees are used. • An angle whose measure is one radian is pictured on a circle whose intercepted arc has length equal to the radius of then circle. • One radian is approximately How many radians is 180°?

6. One complete revolution corresponds to the circumference of the circle as an arc length, so one revolution equals an arc length of • Using a circle of radius 1 (which is called the unit circle) gives us the statement radians. • Some important angles are shown in radians in terms of 1 • Radian measure can be expressed in terms of , but can also be expressed as a decimal. • Angles can be measured in degrees or in radians in terms of , or in radians as a decimal.

7. Label the unit circle with the most commonly used degrees. The angles in standard position whose terminal sides lie on one of the coordinate axes are called Quadrantiles. These common angles in radian form in terms of are used more often in calculus.

8. Measuring Angles in Radians and Degrees What do a pencil, protractor, ruler, and calculator have in common? They are all tools that do not work or think for themselves. They are only as valuable as the person’s mind and ability to use the tools. Solutions? Click again

9. Measuring Angles in Radians and Degrees Be sure to copy the problem and show work to your answer.

10. On the next slide we will link to a website where we can practice our angle knowledge. Move the red dot to 15°. Notice the red arc indicates a positive angle. Consider the angle . In the radian box (below the degree box), type “pi” to see radian measure in terms of . Experiment with positive and negative angles. Take note of the locations for: by moving the red dot to these locations. They will be helpful when you quiz yourself. After experimenting, have students come to the interactive whiteboard and take the quiz. If they miss a problem or if they complete the quiz with all correct answers, they should sit down and another student should come forward, reset the quiz, then try again. You can use this website to check your answers to the problems on the previous slide.

11. Let’s practice! When clicking on the link below, you should see this: Read the directions, then click “Dismiss”. http://www.mathlearning.net/dl2004/Demos/unitCircle.html • Learning Goal: • Students will understand radian measure of an angle, • be able to find co-terminal angles of a given angle; • students will be able to determine the reference angle of a given angle.

12. Finding coterminal angles in degrees or radians Let’s use the website again to verify coterminal angles. Answer the questions first, then verify with the website. Remember, coterminal angles, in standard position, have the same initial side and terminal side, yet have different angle measures. This can happen when considering positive and negative angles. It can also happen as the rotation continues and the angles are greater than What can we add or subtract to find coterminal angles? Name 3 angles coterminal with the given angle. If the angle is given in degrees, name 3 coterminal angles in degrees. If the angle is given in radians, name 3 coterminal angles in radians. http://www.mathlearning.net/dl2004/Demos/unitCircle.html Check you answers:

13. Reference Angles We have defined angles so hat an angle in standard position can have its side lie in any of the 4 quadrants. If Θ is an angle in standard position, its reference angle (Θ prime) is the acute angle formed with the terminal side of Θ and the x-axis. Reference Angle Rules: If is an angle in standard position, its reference angle is the acute angle formed by the terminal side of q the x-axis. The reference angle for any angle ,

14. Let’s determine the reference angle for each of the following. Then, use the website to verify your solutions. http://www.mathlearning.net/dl2004/Demos/unitCircle.html

15. http://www.mathlearning.net/dl2004/Demos/unitCircle.html

16. Learning Goal: • Students will understand radian measure of an angle, • be able to find co-terminal angles of a given angle; • students will be able to determine the reference angle of a given angle. Prior Knowledge Check Return to problems