Math General Angles, Radian Measure, measures of arcs and sectors

Transcription

1 Math General Angles, Radian Measure, measures of arcs and sectors

2 tan h cos? 9 ϴ Tangent ratio gives sides of a right triangle. h h h cos cos

3 cos 3 10 opp 10 sin? 3 ϴ cosine ratio gives sides of a right triangle. h h sin h 91

4 What if the triangle does not have a radius of 1? What is the sine ratio of an angle whose terminal side passes through the point (3, 5)? r =? ϴ 3 5 3, 5 r r r sin sin

5 Now we ll talk about the measure of angles in Degrees and Radians

6 Why 360 º? The idea of dividing a circle into 360 equal pieces dates back to the sexagesimal (60-based) counting system of the ancient Sumarians. Early astronomical calculations linked the sexagesimal system to circles.

7 Angles vs. lengths If I lengthen the sides of the angle, does the measure of the angle change? NO 37 A Notice that the length of the arc depends upon how far the arc is from the vertex of the angle.

8 Radians vs. Degrees How far you are from the vertex of the angle does not change the degree measure of the angle. 37 A Another way of measuring the angle that takes into account the change in the arc length will be useful.

9 Vocabulary Degrees: The measure of an angle as a portion of 360 (the angle measure of a circle) *360 4 Radians: the measure of an angle that takes into account the arc length.

10 Vocabulary Pi: an irrational number that is the ratio of the distance around the circle to the distance across the circle. C D C r C r

11 Vocabulary radian measure of radian a measure circle arc length radius Radian measure: the ratio of the arc length to the distance the arc is from the vertex of the angle. radian radian measure of measure of a circle a circle circumference radius r r radians Units of radians = inches/inches Radian measure has no units! (nice)

12 Vocabulary Degrees: The measure of an angle as a portion of 360 (the angle measure of a circle) *360 4 Radians: the measure of an angle as a portion of π radians (the angle measure of a unit circle) * 4 4

13 Your turn? How many degrees in half a circle? *360 How many radians in half of a circle? * 180 radians

14 radian measure Vocabulary Radian measure: the ratio of the arc length to the radius of the circle: s r arc length radius theta: a Greek letter. Traditionally, we use Greek letters as variables for the measure of an angle. r s s r

15 What is the radian measure of an angle that is ½ of the circle? whole circle radians half circle radians

16 Radian measure: the ratio of the arc length to the radius of the circle: s r A circle has a radius of 5 inches. What is the radian measure of an angle that is ½ of the circle? arc length of the wholecircle r arc length of half the circle r r r* Radian measure of the arc = π radians

17 Your Turn: C r A circle has a radius of mm, what is the circumference of the circle? (leave π in your answer) An arc subtends 1/3 of a circle that has a radius of 3 inches. What is the radian measure of the arc? (leave π in your answer)

18 Your Turn: C r radian measure arc length radius A circle has a circumference of 10π inches. What is the radius of the circle? An angle subtends 1/9 of a circle. The circle has a radius of 10 inches. What is the radian measure of the angle? radian measure 1 9 * (10inches) 10inches 9 radians

19 Problem types you ll see: What is length of the subtended arc? r 3 4 5inches s s r 5* 3 4 s 15 4 inches

20 Degree-Radian Conversion Degree-Radian 180 = π radians 180 To convert radians to degrees, multiply by. radians radians To convert degrees to radians, multiply by. 180 Conversion o 180 These are conversion factors 180 o When you multiply a number by one of these factors, it converts the units.

21 Converting from Degrees to Radian Measure 140 o Converting from Radian Measure to Degrees 180 o 90

22 Your Turn: Convert between radians and degrees. 11 3? o 70?

23 Arc Length s r s r arc length = radius * angle measure (in radians) A circle has a radius of 8 inches. The radian measure of an angle in the circle is What is the length of the arc? s r s 8 inches s s (8 4 radians inches) radians inches

24 Arc Length s r arc length = radius * angle measure (in radians) r = 5 inches Arc length =? radians 3 s (5inches) radians 3 s 5 inches 3

25 Your turn: s r r = inches 60 ( Arc length =? s in)(60 )( ) s s (in)( ) 3 3 in 180

26 Your turn: s r Arc length = 7π inches r 5 inches 14 Angle measure =?

27 Your turn: s r Angle measure = 3π radians Arc length = 1π inches Radius =?

28 Sector Area Problems What is the area of a 0 sector of a circle whose radius is ft? There are two ways to do this problem. 8 ft 0º 1. Find the area of the entire circle then multiply by the fraction of the circle that 0 degrees represents.. A r * (8 ft) * ft ft

29 . Use a formula 8 ft 0º A 1 r What is the area of a 0 sector of a circle whose radius is ft? A A A 0.5(8 ft) (0 Gotcha!!! This formula uses angle measure in radians. 0.5(64ft 1 radians (8 ft) (0 ) 180 ) 9 ) A 3 ft 9

30 Your turn: 10 ft 40º A 9. What is the area of a 40 sector of a circle whose radius is 10 ft? 1 A r A 0.5(10ft) (40 ) A 0.5(100 ft Gotcha!!! This formula uses angle measure in radians. 1 radians (10ft) (40 ) 180 ) 180 A 50 ft 180 A 5 ft 18

31 Your turn: 11. What is the area of a 10º sector of a circle whose radius is 6 ft? 6 ft 10º A 0.5r A 0.5(6 ft) (10 Be careful of the radian/degree gotch. A 0.5(36ft )(10 ) A (18ft ) (18ft ) 1 ft 18 )

32 Your turn: 10. Find the area of a slice of pizza. 14 inch pizza (diameter) Slice is 1/8 of the pizza

33 Your Turn: 11. What is the area of a 45 sector of a circle whose radius = 1 inches? sector area 18 inches 1. What is the radius of a circle with a 100 sector whose sector area is sector area 5 inches radius 3 10 inches 13. The radius of a circle is 0 inches. 00 Sector area inches 9 What is the sector angle? sector angle 0 0