Transcript File

Linear and Angular Speed
 Recall that the formula for velocity is:
 If we want to know angular velocity, then
the formula would be:
𝜃
𝜔=
𝑡
where 𝜃 is the angular distance covered in
radians.
The News Cycle
 A bicycle messenger rides the
bicycle shown.
a) During one delivery, the tires
rotate at a rate of 140
revolutions per minute. Find the angular
speed of the tire in radians per minute.
b) Convert the above angular speed into
linear speed in miles per hour using the
tire diameter provided above.
Another News Cycle
 A bicycle messenger rides the
bicycle shown.
a) Find the tire’s circumference.
b) On part of the trip to the next
delivery, the tire turns at a constant rate of 2.5
revolutions per second. Find the linear speed of
the tire in feet per second (to the thousandths
place), and then miles per hour (to the
hundredths place).
 A. Find the angular speed of this
DVD in radians per second if the
disc rotates at a rate of 3.5
revolutions per second.
 B. If the DVD player overheats and the
disc begins rotating at a slower rate of
3 revolutions per second, find the discs
linear speed in meters per minute.
Staying Warm
 Suppose you are given a ray
on the coordinate plane and
its endpoint P.
 How would you find the
length of the ray r?
 Assume 𝑟 ≠ 0.
 Find all six trigonometric ratios in
terms of x, y, and r.
Unit 4.3: Trigonometric
Functions and the Unit
Circle
Straight to the point
 Find the 6 trig functions for the
following points
 1. (8, -6)
2. (4,3)
3. (-2, -1)
For Future Reference
 If 𝜃 is an angle in standard position, its
reference angle 𝜃′ is the acute angle
formed by the terminal side of 𝜃 and
the x-axis.
 Find the reference angle for the
following angles.
 1.
300𝑜
 4. 240𝑜
2.
2𝜋
−
4
5. 390𝑜
3.
5𝜋
4
 Using the trig ratios from the warm-up, you come
up with the following ratios based on quadrant:
 Using the trig ratios from the warm-up, you come
up with the following ratios based on quadrant:
Today’s Special
 Recall the side lengths
for the two special
right triangles:
 If the length of the
hypotenuse in each
triangle is 1, what is
the length of the legs
of the triangles?
You’re the Investigator
 On the unit circle, the rays will always
have a length of 1 unit (hence the
name).
 Using your knowledge of the special
right triangles, you will be able to find
the points on the unit circle for all of
the multiples of 30𝑜 and 60𝑜
 Reference angles will be useful in
completing the unit circle.
 After you have found the points on the
unit circle, you can then find
𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃 for all multiples of
30𝑜 and 45𝑜 .
 Note that you will always let 𝑐𝑜𝑠𝜃 = _____
and 𝑠𝑖𝑛𝜃 =_____in your equation.
 This leads to 𝑡𝑎𝑛𝜃 = _______
 Being really familiar with the unit circle
is really important.
 It’s pop-quiz important.
 It’s more-than-one-pop-quiz-a-week
important.
 It’s pop-quiz-without-your-notes
important.
 Use your completed
unit circle to find
when each of the
basic trig functions
are positive and when
they are negative.
 This leads to the
acronym that helps
us remember when
each trig function is
positive or negative.
Remember, Remember
 Appalachian State Teaching College
 All Students Take Calculus
 All Students Take Chemistry
 All Silly Tom Cats
 Add Sugar To Coffee
 Or make up your own mnemonic
 Find the exact value of each
expression.
 1.
𝜋
sin( )
3
 3.
𝑡𝑎𝑛270𝑜
𝜋
4
 5. cos( )
 7.
𝑐𝑜𝑡210𝑜
2. 𝑐𝑜𝑠135𝑜
4.
11𝜋
csc( )
6
6. 𝑠𝑖𝑛120𝑜
8.
7𝜋
sec( )
4
 Periodic functions are functions with
values that repeat at regular intervals.
 You can use this periodic nature of
trigonometric functions to find trig
values for angles that don’t fall
between 0 − 360𝑜 using coterminal
angles.
 Find the exact value of each
expression.
 1.
11𝜋
cos( )
4
 3.
19𝜋
tan( )
6
 5.
4𝜋
cos(− )
3
2.
2𝜋
sin(− )
3
4.
13𝜋
sin( )
4
6.
15𝜋
tan( )
6
Buy One, Get 5
 When given a trig ratio and the sign of
a second ratio, find the remaining 5
trig ratios.
 1. 𝑡𝑎𝑛𝜃 = 5/12 and 𝑠𝑖𝑛𝜃 < 0
 2. 𝑠𝑒𝑐𝜃 = 3 and 𝑡𝑎𝑛𝜃 < 0
 3. 𝑠𝑖𝑛𝜃 = 5/7 and 𝑐𝑜𝑡𝜃 > 0
Domo Arigato
 As part of the range of motion
category in a high school
robotics competition, a student
programmed a 20-cm long
robotic arm to pick up an
object at point C and rotate through an
angle of exactly 225𝑜 in order to release it
into a container at point D. Find the
position of the object at point D, relative to
the pivot point O.
 Through Angular Velocity: p238 #34-40 Even
 Through 6 Trig Functions: p251 #2-8 Even
 Through Reference Angle: p251 #18-24 Even
 Through Unit Circle: p251 #10-16 Even and
#26-32 Even
 Through Periodic Nature: p251 #44-50 Even
 Through B1G5: p251 #34-40 Even
 Through Mr. Roboto: p251 #41, 42