Trigonometry, Pt 1: Angles and Their Measure. Mr. Velazquez Honors Precalculus

Transcription

1 Trigonometry, Pt 1: Angles and Their Measure Mr. Velazquez Honors Precalculus

2 Defining Angles An angle is formed by two rays or segments that intersect at a common endpoint. One side of the angle is called the initial side, and the other is called the terminal side. For convenience, it s useful to think of an angle as a stationary initial side with a terminal side that rotates around it, with counterclockwise rotation indicating a positive angle and clockwise rotation indicating a negative angle. An angle is in standard position if: Its vertex is at the origin of the coordinate system Its initial side lies along the positive x-axis

3 Measuring Angles in Degrees

4 Measuring Angles in Radians

5 Measuring Angles in Radians EXAMPLE What is the radian measure θ for an arc of length 15 inches and a radius of 6 inches?

6 Conversion Between Degrees and Radians The arc length of a full circle (360 ) is essentially the entire circumference of the circle. This angle is therefore equal to 2π radians. A half circle has an angle measure equal to π radians.

7 Conversion Between Degrees and Radians EXAMPLE Convert the following angles in degrees into radians. a) 135 b) 45 c) 60 d) 120

8 Conversion Between Degrees and Radians EXAMPLE Convert the following angles in radians into degrees. a) π 2 b) π c) 5π 3 d) π 6

9 Angles in Standard Position Often, we can get a sense of where an angle is located based on certain reference angles. A few of these reference angles are given below:

10 Angles in Standard Position EXAMPLE Draw and label each angle in standard position: a)α = 3π 2 b)β = 2π c) θ = 7π 4

11 Angles in Standard Position Below are select positive and negative angles, given in radians and degrees.

12 A table showing the same standard angle measures and their conversions to radian and degrees. BTW: 1 revolution (equal to 360 or 2π radians) is often used as a unit of angle measurement in science and technology.

13 Coterminal Angles Notice that the angle measurements 90 and 270 both refer to the same exact angle. This means 90 and 270 are coterminal angles. Any angle θ is coterminal with angles of: Where k is any integer. θ + k 360

14 Coterminal Angles EXAMPLE Find a positive angle less than 360 that is coterminal with each of the following: a) 390 b) 405 c) 135

15 Coterminal Angles EXAMPLE Find a positive angle less than 2π radians that is coterminal with each of the following: a) 5π 2 b) 11π 4 c) π 6

16 Length of a Circular Arc EXAMPLE A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 120.

17 Linear and Angular Speed

18 Linear and Angular Speed EXAMPLE A bicycle tire with a radius of 80 cm rotates with an angular speed of 3π radians per second. A piece of gum is stuck to the edge of the tire. What is the linear speed of the piece of gum, in cm/s?

19 Exit Ticket: Angles A windmill is used to generate electricity. Its blades are 12 feet in length, and rotate at an angular speed of 8 revolutions per minute. Find: a) The linear speed at the tips of the blades, in ft/s. b) The central angle (in radians and degrees) each blade will spin through in 3 seconds. Homework: Khan Academy Due 1/17 Remember: Linear Speed v = s t Angular Speed Arc Length ω = θ t s = rθ