Radian and Degree Measures Section 4.1 Radian and Degree Measures

Objective By following instructions students will be able to: Describe angles. Use radian measure. Use degree measure. Use angles to model and solve real-life problems.

Trigonometry Before the development of Calculus, trigonometry was the study of triangle measurements that traditionally dealt with sides and angles. Trigonometry was used in the development of astronomy, navigation, and surveying. After the development of Calculus, the application of Trigonometry expanded and was used to calculate rotations, vibrations, sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.

Why is Trig Important?

What are Angles? Geometry Trigonometry Y axis Terminal side Initial side X axis Vertex Initial side

What are Angles? Positive Angles Negative Angles Counterclockwise Y axis Y axis X axis X axis Def: coterminal – Alpha and Beta are Coterminal because they share the same initial and terminal sides. To find an angle coterminal to a given angle, + or - by a revolution .

Unit Circle Circumference: Suppose radius=1 Then,

Unit Circle Circumference: Suppose radius=1 Then,

Unit Circle Circumference: Suppose radius=1 Then,

Unit Circle Circumference: Suppose radius=1 Then,

Example 1: Sketch the angle. Find a positive and negative co-terminal. a) b) c)

Complimentary Angles Supplementary Angles Two angles whose measures add up to 90 degrees. Two angles whose measures add up to 180 degrees.

Example 2: Complementary and Supplementary Angles If possible, find the complement and supplement of a) b)

Degrees Radians RadiansDegrees

Example 3: Converting Degrees to Radians Express each angle in radian measure. a) b)

Example 4: Converting Radians to Degrees Express each angle in degrees. a) b)

Unit Circle Circumference: Suppose radius=1 Then,

Unit Circle Circumference: Suppose radius=1 Then,

Arc Length Recall: By solving for the arc length, Where is measured in radians. .

Example 5: Finding Arc Length A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240 degrees.

Linear Speed Angular Speed The formula for arc length can be used to analyze the motion of a particle moving at a constant speed. Linear Speed Angular Speed Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is: If is the angle (in radian measure) corresponding to the arc length s, the angular speed of the particle is:

Example 6: Finding Linear Speed The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand. 10.2 cm

Example 7: Finding Angular and Linear Speed A 10 inch radius lawn roller makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second. Find the speed of the tractor that is pulling the roller.

Revisit Objective Did we… Describe angles? Use radian measure? Use degree measure? Use angles to model and solve real-life problems?

Homework HW: I: pg 187 #s 18, 28 pg 209 # 2 II: pg 291 #5-17 ODD, 27-35 ODD, 43-65 ODD III: pg 294 #s 99, 100, 103