Unit 6 Introduction to Trigonometry Degrees and Radians (Unit 6.2)

Unit 6 Introduction to Trigonometr Degrees and Radians (Unit 6.2) William (Bill) Finch Mathematics Department Denton High School

Lesson Goals When ou have completed this lesson ou will: Understand an angle as a measure of rotation. Understand radian and degree measures. Be able to convert between radian and degree measure. Be able to calculate arc length and sector area. Be able to find angular and linear speeds. Radian/Degree 2 / 35

Angles in Standard Position An angle in standard position: starts on positive -ais (initial side) rotates counter-clockwise for positive angles rotates clockwise for negative angles often named with Greek letters theta... θ alpha... α beta... β Terminal Terminal Negative Positive Initial Initial Radian/Degree 3 / 35

Degree Measure 135 120 90 60 45 150 30 180 360 0 (0 ) 210 330 225 315 240 270 300 Radian/Degree 4 / 35

Degree-Minutes-Seconds (DMS) A fraction of a degree can be epressed as a decimal fraction, but historicall the degree was divided into minutes ( ) and seconds ( ). 1 = 60 and 1 = 60 For eample, 32.125 = 32 7 30 Read 32 degrees, 7 minutes, and 30 seconds. Radian/Degree 5 / 35

Eample 1 Convert to decimal degrees. a) 25 15 b) 12 10 33 Radian/Degree 6 / 35

Calculator Instructions TI-84 Radian/Degree 7 / 35

Eample 2 Convert to degree-minutes-seconds. a) 48.4 b) 21.456 Radian/Degree 8 / 35

Calculator Instructions TI-84 Radian/Degree 9 / 35

Radian Measure One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle: θ = s r where θ is measured in radians. r θ r s Note that in the diagram above the radius r of the circle is the same length as the arc s intercepted b the two radii, so θ = 1 rad when s = r. Radian/Degree 10 / 35

Radian Measure The circumference of a circle is one revolution around the circle. C = 2πr s = 2πr s r = 2π θ = 2π θ 6.28 A central angle θ that is one revolution is 2π radians. θ Radian/Degree 11 / 35

Radian Measure One revolution around a circle is slightl more than 6 radians. 2 rad 3 rad 4 rad r 1 rad s = r 6 rad 5 rad Radian/Degree 12 / 35

Radian Measure 150 135 120 3π 4 5π 6 2π 3 90 π 2 π 3 60 π 4 π 6 45 30 180 π 2π 360 0 210 7π 6 5π 4 4π 3 3π 2 5π 3 11π 6 7π 4 225 315 240 270 300 330 Radian/Degree 13 / 35

Special Angles Learn Them! π 2 π 2π 3π 2 Radian/Degree 14 / 35

Special Angles Learn Them! π 4 3π 4 5π 4 7π 4 Radian/Degree 15 / 35

Special Angles Learn Them! π 3 2π 3 4π 3 5π 3 Radian/Degree 16 / 35

Special Angles Learn Them! π 6 5π 6 7π 6 11π 6 Radian/Degree 17 / 35

Radian Measure Quadrant II π 2 < θ < π (obtuse angles) θ = π 2 Quadrant I 0 < θ < π 2 (acute angles) θ = π θ = 0 Quadrant III π < θ < 3π 2 Quadrant IV 3π 2 < θ < 2π θ = 3π 2 Radian/Degree 18 / 35

Radian-Degree Conversion Set up and solve this proportion: radian degree = π rad 180 Hint alwas set up the proportion with the unknown angle measure in the numerator. Radian/Degree 19 / 35

Eample 3 Convert to radian measure. a) 120 b) 30 Radian/Degree 20 / 35

Eample 4 Convert to degree measure. a) 3π 4 b) 3π 2 Radian/Degree 21 / 35

Coterminal Angles Coterminal angles have the same initial and terminal sides. α β β α To find a coterminal angle to some angle θ either add or subtract a multiple of 2π (or 360 ): θ ± n 2π θ ± n 360 Radian/Degree 22 / 35

Eample 5 Sketch the angle given (in radians): θ = 2π 3 Then find two coterminal angles: one positive and one negative. Radian/Degree 23 / 35

Eample 6 Sketch the angle given (in radians): α = π 4 Then find two coterminal angles: one positive and one negative. Radian/Degree 24 / 35

Eample 7 Sketch the angle given (in degrees): β = 25 Then find two coterminal angles: one positive and one negative. Radian/Degree 25 / 35

Eample 8 Sketch the angle given (in degrees): θ = 150 Then find two coterminal angles: one positive and one negative. Radian/Degree 26 / 35

Arc Length The relationship between a central angle and the length of the intercepted arc is where θ is in radians. s = rθ s θ r Radian/Degree 27 / 35

Eample 9 A circle has a radius of 5 inches. Find the length of the arc intercepted b a central angle of 120. Radian/Degree 28 / 35

Eample 10 Winnipeg, Manitoba (Canada) is approimatel due north of Dallas. Winnipeg is at a latitude of 49 53 0 N, and Dallas is at a latitude of 32 47 39 N. Use the given information to find the distance between Winnipeg and Dallas (assume the Earth is a perfect sphere with a radius of 4000 miles). Radian/Degree 29 / 35

Area of a Sector A sector of a circle is the region bounded b two radii and their intercepted arc. θ r The area of a sector is A = 1 2 r 2 θ (where θ is in radians). Radian/Degree 30 / 35

Eample 11 A sector has a radius of 12 inches and a central angle of 100. Find the area of the sector. Radian/Degree 31 / 35

Eample 12 Find the approimate area swept b the wiper blade shown, if the total length of the windshield wiper mechanism is 26 inches. Radian/Degree 32 / 35

Linear and Angular Speed An object moving along an arc has a linear speed given b s ν = arc length time = s t θ r An object moving along an arc has an angular speed given b ω = central angle time = θ t Radian/Degree 33 / 35

Introduction Angles Degrees Angles Radians Coterminal Applications Summar Eample 13 A biccle wheel has a radius of 35 cm. A chalk mark is made on the tire and then the tire is spun completing one full revolution in 0.8 seconds. a) Determine the linear speed of the chalk mark. b) Determine the angular speed. W. Finch Radian/Degree DHS Math Dept 34 / 35

What You Learned You can now: Understand an angle as a measure of rotation. Understand radian and degree measures. Be able to convert between radian and degree measure. Be able to calculate arc length and sector area. Be able to find angular and linear speeds. Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35, 39, 41, 43, 45, 51, 55, 57, 59 Radian/Degree 35 / 35