Radian and Degree Measure Algebra III, Sec. 4.1 Objective Describe an angle. Convert between radian and degree measure.

Important Vocabulary Trigonometry – Central angle of a circle – Complementary angles – Supplementary angles – Degree –

Angles An angle is determined by… rotating a ray about its endpoint terminal side vertex initial side

Angles An angle is in standard position when… the vertex is at the origin and the initial side coincides with the positive x-axis A positive angle is generated by a _________________ rotation; whereas a negative angle is generated by a ________________ rotation. If two angles are coterminal, then they have… the same initial and terminal sides. counterclockwise clockwise

Clockwise MOVEMENT GIVES A NEGATIVE ANGLE -60°

Counterclockwise MOVEMENT GIVES A POSITIVE ANGLE 60°

REMEMBER ANGLE MEASURE IS ALWAYS POSITIVE NEGATIVE TELLS DIRECTION

Coterminal Angles ANGLES THAT SHARE THE SAME TERMINAL SIDE MUST DIFFER BY WHOLE ROTATION ± 360°

NAME 3 POSITIVE ANGLES THAT ARE COTERMINAL WITH 30° 30 + 360 = 390° 30° 390 + 360 = 750° 750 + 360 = 1110° HINT: TO GET BACK TO SAME PLACE YOU MUST GO ALL THE WAY AROUND IN THE POSITIVE DIRECTION OR NEGATIVE DIRECTION

NAME 3 NEGATIVE ANGLES THAT WOULD BE COTERMINAL WITH 30° 30 - 360 = -330° -330 - 360 = -690° 30° -690 - 360 = -1050° HINT: TO GET BACK TO SAME PLACE YOU MUST GO ALL THE WAY AROUND IN THE POSITIVE DIRECTION OR NEGATIVE DIRECTION

Q III I II IV II IV III II 220° 400° 900° -200° -750° 110° EXAMPLE: Tell what quadrant each angle would lie in or state that it is quadrantal Q 220° 400° 900° -200° -750° 110° 650° 1200° -120° III I II IV II IV III II

Radian Measure The measure of an angle is determined by… the amount of rotation from the initial side to the terminal side. One radian is the measure of a central angle that… intercepts an arc s equal in length to the radius r of the circle.

Radian Measure A full revolution around a circle of radius r corresponds to an angle of ____________ radians. A half revolution around a circle of radius r corresponds to an angle of ___________ radians. Angles with measures between 0 and π/2 radians are ___________ angles. Angles with measures between π/2 and π radians are ___________ angles. 2π π acute obtuse

Example 1 Find the negative and positive coterminal angles of… a. b. c.

Example 2 Find the complement and supplement of the angle… a. b. no complement

Examples (on your handout) Find an angle that is coterminal with θ = - π/8 Find the supplement of θ = π/4

Degree Measure A full revolution (counterclockwise) around a circle corresponds to __________ degrees. A half revolution around a circle corresponds to __________ degrees. To convert degrees to radians… multiply by To convert radians to degrees… 360 180

Example 3 Express each angle in radian measure as a multiple of π (do not use a calculator). a. 420° b. 280° c. -30° radians radians radians

Example 4 Express each angle in degree measure (do not use a calculator). a. b. c. 3 radians

Examples (on your handout) Convert 120° to radians. Convert 9π/8 radians to degrees.

Examples (on your handout) Complete the table… θ (degrees) 0° 45° 90° 270° θ (radians) π/6 π/3 π 2π θ (degrees) 0° 30° 45° 60° 90° 180° 270° 360° θ (radians) π/6 π/4 π/3 π/2 π 3π/2 2π

Applications

Arc Length on a Circle s=rθ (θ in radians)

Example 5 A circle has a radius of 10 inches. Find the length of the arc intercepted by a central angle of 140°. About 24.43 inches

Linear Speed (v) Where s is the arc length traced by point P at time t

Angular Speed (ω) Measure of how fast an angle is changing θ in radians ω in radians per unit of time Relates linear and angular speed.

Example 6 The second hand of a watch is 1.3 cm long. Find the linear speed of the tip of this second hand as it passes around the watch face. About 0.14 cm/s

Example 7 The circular blade on a saw rotates at 4200 revolutions per minute. Find the angular speed in radians per second. The blade has a radius of 6 inches. Find the linear speed of a blade tip in inches per second. 140π radians per second About 2639 inches per second

Area of a Sector of a Circle The area A of a sector of a circle of radius r and central angle θ is given by θ in radians.

Example 8 A sprinkler on a golf course is set to spray water over a distance of 75 feet and rotates through an angle of 135°. Find the area of the fairway watered by the sprinkler. 6627 square feet

Example (on your handout) A 6-inch diameter gear makes 2.5 revolutions per second. Find the angular speed of the gear in radians per second.