6.1: Angle Measure in degrees

Transcription

1 6.1: Angle Measure in degrees

2 How to measure angles Numbers on protractor = angle measure in degrees 1 full rotation = 360 degrees = 360 half rotation = quarter rotation = 1/8 rotation = 1 =

3 Right angle and Straight angle

4 Acute angle and obtuse angle

5 Coterminal angles Angles that share the same initial and terminal sides are called coterminal angles.

6 List a few coterminal angles of the following angles

7 Special Triangles

8 Examples

9 Radian measure of an angle Here we have a central circle of radius r. Definition (central angle) A central angle is an angle whose vertex is at the center of the circle. Definition (Radian measure of an angle) If θ is a central angle in a circle of radius r, and is subtended by an arc length of s, then θ = s r

10 Examples

11 Radian measure of an angle (Continued) So it s a new way of defining angle measures. In words, Definition Radian is the ratio between an arc length and its radius. That is, Radian = arc length radius (or, θ = s r ) Then, 1 radian is when arc length = radius 2 radian is when arc length = 2 times the radius etc...

12 Examples Let the angle measure starts at P. Example 1. r = 8cm, s = 4cm, what is θ in radians? Example 2. r = 8cm, s = 8cm, what is θ in radians? Example 3. r = 8cm, s = 16cm, what is θ in radians?

13 Special case: when we have a unit circle (i.e. r = 1 ) arc length Radian = radius Thus, for the unit circle with r = 1, 1 radian is when arc length = 1 2 radian is when arc length = 2 etc... = arc length.

14 Remember π ? What is π? Definition π is the ratio of the circumference (c) to its diameter (2r). π = c 2r Thus, for general r, c = If r = 1, then what is c? What does this mean?

15 2π = 360 What is 1 radian in terms of degrees? What is 1 degree in terms of radians?

16 Special angles Convert the following angles in degrees to radians. 0 = 30 = 45 = 60 = 90 = 180 = 270 = 360 =

17 Special angles (Continued) Convert the following angles in radians to degrees. 0 = π 6 = π 4 = π 3 = π 2 =

18 More problems Convert from the degree measures to radians (and vice versa). θ = 210 θ = 2π 3 θ = 3.7

19 Arc Length (s) Idea: Given a circle, an arc length can be figured out as long as I know the radius of the circle and the angle it subtends to. Recall that θ in radians is given by the following: θ = s r where s = arc length and r = radius. Then, s = rθ Example 1. r = 10cm. Find s subtended by an angle of 3.5 rad.

20 Angular and Linear Velocity Definition (Angular velocity) The angular velocity (ω) is the amount of rotation per unit time. That is, ω = θ t Ex 1) The round-a-bout makes 4/3 revolutions per 1 second. What is the angular velocity ω? Ex 2) The round-a-bout makes 1 revolution in 2 seconds. ω?

21 Linear Velocity Definition (Linear velocity) The linear velocity (V ) is the distance traveled by a point on the circumference per unit time. That is, V = s t = rθ t = rω Ex) Point P is rotating around the circumference of a circle with r = 2ft at a constant rate. If it takes 4 seconds for P to rotate through an angle of 360, what is the linear velocity of P?

22 Example The wheels on a racing bicycle have a radius of 13 in. How fast is the cyclist traveling in miles per hour (mph) if the wheels are turning at 300 rotations per minute (rpm)? (Note: 1 mile = 5280 ft = 5280 x 12 inches, and 1 min = 1/60 hr)

23 The area (A) of a circular sector Recall that The area of a circle of radius r is given by πr 2. one full revolution = 2π radians Then, note that A total area of circle = angle subtended by arc s angle subtended by circumference = θ 2π A = = 1 2 r 2 θ

24 Example In the unit circle, what is the area of the quarter of the circle? half the circle? full circle? (Note multiple ways)

25 Example - Area of a slice of pizza What is the area of one slice of the following pizza? The whole pizza, whose diameter is 14 inches, was cut equally into 6 pieces.

26 Review of Ch 6.1 Definition Radian is the ratio between an arc length and its radius. That is, Radian = arc length radius (or, θ = s r ) From the above definition, we also have s = rθ. Since 2π = 360 ( or π = 180 ) To convert from degree measures to radian, multiply π 180. To convert from radian measures to degree, multiply 180 π.

27 Review of Ch 6.1 (Continued) Given a right triangle ABC with θ as follows,, the ratios of the three sides are; If θ = 30 (or π 6 ) b : a : c = 3 : 1 : 2 If θ = 45 (or π 4 ) b : a : c = 1 : 1 : 2 If θ = 60 (or π 3 ) b : a : c = 1 : 3 : 2

28 Review of Ch 6.1 (Continued) Definition (Angular velocity vs. Linear velocity) The angular velocity (ω) is the amount of rotation per unit time: ω = θ t The linear velocity (V ) is the distance traveled by a point on the circumference per unit time: V = s t = rθ t = rω Also, area of a sector of a circle: A = 1 2 r 2 θ

29 HW for Ch. 6.1 Note: Show all work. 1, 8, 12, 45, 46, 58, 64, 81, 83, 87