Objective: To distinguish between degree and radian measure, and to solve problems using both.

Transcription

1 CHAPTER 3 LESSON 1 Teacher s Guide Radian Measure AW 3.2 MP 4.1 Objective: To distinguish between degree and radian measure, and to solve problems using both. Prerequisites Define the following concepts. Circle A circle is the set of all points in a plane that are a given distance from a given point in the same plane. The given point is called the centre of the circle. Radius The radius is a segment from a point of a circle to the centre. The length of this segment is also called the radius Circumference The circumference is the distance around a circle. Arc of a circle An arc is that part of a circle that consists of 2 points on the circle and the continuous part of the circle between the 2 points.

2 Central angle A central angle is an angle whose vertex is at the centre of a circle and whose sides contain radii of the circle. Sector of a circle A sector is the region enclosed by an arc and 2 radii that intersect the endpoints of the arc. Notes: The shaded area is a sector. θ is a central angle. r is the radius. The distance a is called an arc length. Developing Radian Measure 2 Questions: 1. What is the circumference of a circle with radius r? 2. How many degrees are there in a complete rotation? Fill in the following table: Answer: 2r Answer: 360º Radius Circumference Number of times the radius fits around the circumference 1 2 2/1 = /2 = /3 = 2 r 2r 2r/r = 2 What can you conclude about the relationship between the radius and the number of times the radius can wrap around the circle? No matter what the radius is, it will wrap around the circle exactly 2 times.

3 Graphically, the radius will wrap around the circle as follows. (The radius is r. ) 1r + 1r + 1r + 1r + 1r + 1r r = r = 2r If the radius varies in size, will the corresponding central angle measurements change or stay the same? Why? The size of the central angles will remain the same. No matter what the size of the radius, the pizza will be split into the same number of pieces. Radians are related to the radius. According to the diagram below, we can see the relationship between the radius, arc length, and 1 radian. An arc of length r of a circle of radius r subtends an angle of 1 radian. In other words, 1 radian is the measure of a central angle that is subtended by an arc equal in length to the radius of the circle.

4 Developing the Relationship Between Radians and Degrees 1. How many degrees does it take to go around a circle? Answer: 360º 2. How many times does the radius go around a circle? Answer: 2 Both measurements can take us completely around the circle. Complete the following relationships: Going around the circle in degrees = Going around the circle in radians 360 = 2 radians Converting Radians to Degrees If 2 radians = 360, then radian = = In general, 180 x radians = x degrees Converting Degrees to Radians If 360 = 2 radians, then 2 1 = 360 radians = 180 radians In general, x = x radians 180 What is the size of 1 radian in degrees? 1 radian = 180 = 180 = Note: Any angle measurement given without a unit is assumed to be in radians. E.g. θ = 2 means θ = 2 radians.

5 Convert each of the following to radians or degrees. (a) 30º = 6 (e) 225º = 5 4 (i) 7 6 = 210º (b) 2 = 90º (f) 4 = 45º (j) 45º = 4 (c) 2 3 = 120º (g) 2 radians = º (k) 3 5 = 108º (d) 36º = 5 (h) 252º = 7 5 (l) 7 radians = º Finding Arc Length Consider a circle with radius r, and an arc of length a that subtends a central angle θ. Both the arc length and the central angle subtended by the arc are parts of the whole circle in some way. The arc length is a part of the total circumference of the circle. The central angle subtended by the arc is part of the total rotational angle.

6 Write the proportion by following the diagram and filling in the blanks: a θ r Note: The angle θ is measured in radians. arc length total circumference = central angle measure total angle measure of 1 rotation a 2r = θ 2 By solving for a in the relationship, the equation for calculating the arc length is: a = rθ (where the angle θ is in radians) If we set up the same relationship but with the central angle in degrees, we get: arc length total circumference = central angle measure total angle measure of 1 rotation a θ = 2r 360 a = o 180 r θ (where the angle θ is in degrees) Note how much simpler the arc length formula is if we use radian measure for the central angle.

7 Example 1: Determine the arc length in a circle of radius 10 cm if: (a) the central angle is 5 radians a = rθ a = 10 5 = 50 cm b) the central angle is 25º. Example 2: a = r 180 θ a = 10 cm = 4.36 cm Determine the central angle (in radians) subtended by an arc of length 3 cm in a circle of radius 10cm. a = rθ θ = a r = 3 cm 10 cm = 0.3