Use your notes from last week: 1
Use your notes from last week: • Find the value of x and y. A) 1 y 45° x Use your notes from last week: • Find the value of x and y. B) 1 30° x y Use your notes from last week: • Find the value of x and y. c) 1 60° x y Section 7-1 Measurement of Angles Objective: To find the measure of an angle in either degrees or radians and to find coterminal angles. What we are going to learn in Sec 7.1 • • • • • Vocabulary Angle measure in degrees and radians Standard position The critical values on the Unit Circle Coterminal angles Section 7-1 Measurement of Angles • Objective: 1. To find the measure of an angle in either degrees or radians. 2. To find coterminal angles. Common Terms • Initial ray is the ray that an angle starts from. • Terminal ray is the ray that an angle ends on. • Vertex Common Terms • A revolution is one complete circular motion. Angles in standard position y x Standard Position Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248 Vertex at origin Copyright © Houghton Mifflin Company. All rights reserved. The initial side Digital Figures, 4–3 of an angle in standard position is always located on the positive x-axis. Angles in standard position • • • • The vertex of the angle is on (0,0). Initial ray starts on the positive x-axis The angle is measure counter clockwise. The terminal ray can be in any of the quadrants. Angle describes the amount and direction of rotation 120° –210° Positive Angle: rotates counter-clockwise (CCW) Negative Angle: rotates clockwise (CW) PositiveSection and negative angles4.3, Positive and 4.1, Figure Negative Angles, pg. 248 When sketching angles, always use an arrow to show direction. Copyright © Houghton Mifflin Company. All rights reserved. 13 Digital Figures, 4–4 Units of angle measurement • There are two ways to measure an angle: Degrees & Radians Units of angle measurement • There are two ways to measure an angle: Degree: 1/360th of a circle. That is the measure one sees on a protractor and most people are familiar with. Angles can be further split into 60 minutes per degree and 60 seconds per minute. Quadrantal Angle • If the terminal ray of an angle in standard position lies along an axis the angle is called a quadrantal angle. • The measure of a quadrantal angle is 𝜋 always a multiple of 90°, or 2 Quadrantal angles Standard Position • When an angle is shown in a coordinate plane, it usually appears in standard position, with its vertex at the origin and its initial ray along the positive xaxis. Degrees On one of the circles provided measure 1° Radian Measure • Use the string provided to measure the radius. • Start on the “x-axis” and use the string to measure an arc the same length on the circle. • The angle created is one radian. Angle θ is one radian Units of angle measurement Radian: when the arc of circle has the same length as the radius of the circle. Angle a measures 1 radian. Arc Le nght=radius 1 radian Radius approximations • 1 radian ~ 57.2958 degrees • 1 degree ~ 0.0174533 radians* • *note: the radian measure is usually stated as a fraction of . Sec 7.1 day 2 Warm up • While I check your UC, work on the following: a) Display the measure of one radian on circle. Display the measure of two radians on a cirlce. b) Describe what one radian is in terms of the radius of a circle r. c) Draw a circle and identify a central angle. Describe relationship between central angle and the intercepting arc. Find the measure of the central angle 𝟏 𝟐 The central angle shown has a measure of radian. What is the length of arc 𝑪𝑫? Find the measure of the central angle COH Length CGH = 4 cm Measure of central angle: • For radian measure: s r s= arc length r= radius For degree measure: 180 s r Section 4.1, Figure 4.7, Common Radian Measure Radian Angles, pg. 249 30 Working with Radians 1 180 180° 1 𝑟𝑎𝑑𝑖𝑎𝑛 = 𝜋 The conversion process • When converting between the two units of angle measurement, start with the following template: 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛 = 180° 𝐷𝑒𝑔𝑟𝑒𝑒 Example 1 Convert 196° to radians 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛 = 180° 𝐷𝑒𝑔𝑟𝑒𝑒 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛 = 180° 196° Radian measure 196 is= 𝜋 180 = 49 𝜋 45 Convert 2 𝜋 3 to degrees. 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛 = 180° 𝐷𝑒𝑔𝑟𝑒𝑒 2 𝜋 𝜋 3 = 180° 𝑥 2 180° × 𝜋 3 = 120° 𝑥= 𝜋 Calculator • 2nd APP (Angle) • Use DMS to convert to Degree, minute and second. • Use Angle to change 40° 20’ to a decimal value. • For more information click here Coterminal angles • Two angles in standard position are called coterminal angles if they have the same terminal ray. • For any given angle there infinitely many coterminal angles. Example • Find two angles, one positive and one negative, that are coterminal with the angle 52°. Sketch all three angles Solution • 52 + 360 = 412 • 52 + 360 × 2 = 772 • 52 + 360 × 3 = 1132 • 52 − 360 = −308 • 52 − 360 × 2 = −668 • 52 − 360 × 3 = −1028 Example • Find two angles, one positive and one negative, that are coterminal with the angle 4 •Sketch all three angles. y Coterminal Angles generalized: • Degree measure: θ 360°n • Radian measure: θ 2π n • Where n is a counting number. Helpful websites • Trig flash cards • http://mathmistakes.info/facts/TrigFacts/ • Hot math flash cards: • http://hotmath.com/learning_activities/inter activities/trig_flashcard.swf Homework • Sec 7.1 Written Exercises • Problems # 1-8 all and • # 9-29 odds • UC with coordinates filled out. 1 degree = 60 minutes 1 minute = 60 seconds 1° = 60 1 = 60 3600 So … 1 degree = _________seconds Express 365010as decimal degrees 36 50 60 10 3600 36 + .8333 + .00277 36.8361 OR Use your calculator!! Express 365010as decimal degrees Enter 36 Press this button ’ ’’ Press enter Enter 50 Press this button ’ ’’ Go over to the ’ symbol -- enter Enter 10 Press this button ’ ’’ Go over to the ’’ symbol -- enter Press enter Convert 50 47’ 50’’ to decimal degree 50.7972 Convert 125 27’ 6’’ to decimal degree 125.4517 Can you go backwards and convert the decimal degree to degrees minutes seconds? Enter 125.4517 Go to DMS hit enter. Express 50.525 in degrees, minutes, seconds 50º + .525(60) 50º + 36.5 50º + 36 + .5(60) 50 degrees, 36 minutes, 30 seconds
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