Angles in a Unit Circle PDF

# 2nd QUARTER - WEEK 1 ## 3-HOUR SESSION # ANGLES IN A UNIT CIRCLE ## MEASURES OF ARCS IN A UNIT CIRCLE ### STEM PC11T-IIa-1; STEM PC11T-IIa-2; STEM PC11T-IIa-3 # What I Need to Know - OBJECTIVES - Illustrate the unit circle and the relationship between the linear and angular measures of arcs in a...

# 2nd QUARTER - WEEK 1 ## 3-HOUR SESSION # ANGLES IN A UNIT CIRCLE ## MEASURES OF ARCS IN A UNIT CIRCLE ### STEM PC11T-IIa-1; STEM PC11T-IIa-2; STEM PC11T-IIa-3 # What I Need to Know - OBJECTIVES - Illustrate the unit circle and the relationship between the linear and angular measures of arcs in a unit circle. - Convert degree measure to radian measure, and vice versa. - Illustrate angles in standard position and coterminal angles. # What's In - REVIEW ## ANGLE MEASURE An angle is formed by rotating a ray about its endpoint. An angle in geometry is defined as a union of rays (that is, static). An angle in trigonometry is a rotation of a ray, and, therefore, has no limit. - It has an initial side and a terminal side. - It has positive and negative directions and measures. In the figure shown below, the initial side of ∠AOB is OA, while its terminal side is OB. An angle is said to be positive if the ray rotates in a counterclockwise direction, and the angle is negative if it rotates in a clockwise direction. Why would you even have negative angles? An angle is in standard position if it is drawn in the xy-plane with: - vertex at the origin (0, 0) and - its initial side on the positive x-axis. There are three units with which to measure angles: revolutions, degrees, and radians. One revolution is the measure of an angle that rotates until its terminal side meets its initial side which is equal to 360°. To measure angles in degrees, we use degrees, minutes, seconds. A central angle of a circle measures one degree, written 1°, if it intercepts 1/360 of the circumference of the circle. One minute, written 1', is 1/60 of 1°, while one second, written 1", is 1/60 of 1'. Radian measure allows us to treat the trigonometric functions as functions with the set of real numbers as domains, rather than angles. A central angle of the unit circle that intercepts an arc of the circle with length 1 unit is said to have a measure of one radian, written 1 rad. # What's New - ICE BREAKER ## ANGLE MEASURE A unit circle has circumference 2π, a central angle that measures 360° has measure equivalent to 2π radians. Thus, we obtain the following conversion rules. - To convert a degree measure to radian, multiply it by π/180. - To convert a radian measure to degree, multiply it by 180/π. A diagram of a unit circle shows multiple angles in degrees and radians. 0° = 0; 360° = 2π 30° = π/6 45° = π/4 60° = π/3 90° = π/2 120° = 2π/3 135° = 3π/4 150° = 5π/6 180° = π 210° = 7π/6 225° = 5π/4 240° = 4π/3 270° = 3π/2 300° = 5π/3 315° = 7π/4 330° = 11π/6 ## EXAMPLE: 1. Convert the following degree measures to radian measure. - 60° - 90° - 150° 2. Convert the following radian measures to degree measure. - π/9 rad - 3π/4 rad # What is It - DISCUSSION ## COTERMINAL ANGLES: Angles in standard position whose terminal sides coincide are called coterminal angles. - These angles are not necessarily equal. - Degree measures of coterminal angles differ by multiples of 360°. A diagram shows angles in standard position that are coterminal. - 45° - -315° - 405° ## EXAMPLE: 1. Find the angle between 0° and 360° (if in degrees) or between 0 rad and 2π rad (if in radians) that is coterminal with the given angle. - 736° - -28°48′65″ - 13π/2 rad - 10 rad 2. Find the angle between -360° and 0° (if in degrees) or between -2π rad and 0 rad (if in radians) that is coterminal with the given angle. - 142° - -400°1′23″ - π/8 rad - -20 rad ## REFERENCE ANGLES An angle's reference angle is the angle formed by the its terminal side and the x-axis. Measures from 0° to 90°. A diagram of a unit circle shows multiple angles in different quadrants, all with an angle 't', and the reference angle to 't' for each quadrant. # What's More - PRACTICE ## ARC LENGTH AND AREA OF A SECTOR An arc is a portion of the circumference of a circle while a sector is a part of a circle made of the arc along with its two radii. In a circle of radius r, the length of an arc s intercepted by a central angle with measure θ in radians is given by s = rθ. In a circle of radius r, the area A of a sector with a central angle measuring θ radians is: A = 1/2 θr^2 ## EXAMPLE 1. In a circle of radius 7 feet, find the length of the arc that subtends a central angle of 5 radians. 2. A central angle θ in a circle of radius 20 m is subtended by an arc of length 15π m. Find the measure of θ in degrees. 3. Find the area of a sector of a circle with central angle that measures 75° if the radius of the circle is 6 m. # What I Have Learned - SUMMARY # Assessment - EVALUATION # Additional Activities - ASSIGNMENT