Trigonometry “Measurement of triangles”

Initial side Angle Terminal side Vertex Angles are always labeled with either a capital letter or a Greek letter When the initial side is on the x axis, the angle is in STANDARD POSITION

Types of angles: Positive angles: Negative angles: Coterminal angles: Generated by a counterclockwise rotation Generated by a clockwise rotation Angles that have the same initial and terminal sides There are pictures on page 262

Measures of angles: determined by the amount of rotation from the initial side to the terminal side One way to measure are: RADIANS Measure of a central angle O that intercepts an arc s equal in length to the radians r of the circle

Radians… Because the circumference of a circle is units it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length: A radian measure of a central angle is

Because the radian measure of an angle of one full revolution is …

You need to memorize!! Q2 Q1 Q3 Q4

What can you tell me about and ? Coterminal Angles To find coterminal angles: Practice finding coterminal angles:

Complementary and Supplementary Angles Complementary: sum = Supplementary: sum = Lets find the complement and supplement of

Day 1 HW pg. 269 # 1 – 23 odd

Warm Up: Complete the following chart with the correct radian measure Complete the following problems: pg. 269 #’s 4, 6, 8, 12, 14, 18

Degree Measure The second way to measure angles. What is the relationship between degrees and radians?

Fill in the following circle with the correct radian and degree measures:

If NO unit measures are specified, then radian measure is IMPLIED, so assume radians!! Let’s practice converting:

Another way to express degrees is to denote fractional parts of degrees by minutes (‘) and seconds(”). ??? How would represent 64 degrees, 32 minutes, and 47 seconds in this form?

Convert this angle to decimal degree form:

To convert from decimal degree form to form, you can use your calculator. 1. Type in decimal degree 2. 2 nd angle (apps) 3. Choose 4 (DMS) 4. Enter Try: 1.) o 2.) o

Day 2 HW pg. 269 #’s 24 – 77 by 3’s

APPLICATIONS Finding arc length: Recall that, therefore s = ? *always in radians Let’s do example 5 on page 267. s = 2 πr Will be helpful to think of θ: 2 π times the number of revolutions

Some more practice: 1. Find the angle measure of an angle with a radius of 10 and arc length of On a circle with a radius of 9 inches, find the length of the arc intercepted by a central angle of 140 degrees.

Arc length is used to analyze motion of a particle moving at a constant speed along a circular path. Linear speed : Measures how fast the particle moves Angular speed: Measures how fast the angle changes

Practice finding linear speed… The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of this second hand as it passes around the clock face.

Some practice… 1. A lawn roller with a 20 inch radius makes 2 revolutions per second. Find the linear speed in inches per second of a point on the outside of the roller after 1 second. 2. A bicycle wheel is turning at a rate of 15 revolutions per minute. The wheel is 24 inches in diameter. Find the linear velocity in feet per second of a piece of gum stuck to the outside of the wheel in/sec 1.57 ft/sec

Practice finding angular speed and linear speed: A lawn roller with a 10 inch radius makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second, then find the speed of the tractor that is pulling the roller (linear speed).

Some practice: 1. The wheel of a truck is turning at a rate of 6 revolutions per second. The wheel is 4 feet in diameter. a.) Find the angular speed in radians per second. b.) Find the linear speed, in feet per second, of a point on the rim of the wheel. 12π rad/sec 75.4 ft/sec

HW – Day 3 pg. 269 # 25, 29, 35, 37, 41, 49, 51, 53, 55, 57, 63, 67, 79, 81, 83, 87, 99

Day 4 Warm Up pg. 271 # 99 Day 4 – With a partner pg. 271 # 91, 95, 96, 97, 98, 100, 102

Study problems for quiz… Pg. 269 # 8, 14, 18, 22, 30, 38, 42, 58, 64, 68, 80, 84, 88