Monday, March 18, 2013


Midterm: Wednesday
Topic: Periodic Functions & Trig
Aim: How can we graph from a function rule?
Do Now: What is the amplitude of the function y = ½ sin 2Θ?
Graphing from a function rule
y = 2 sin 2Θ
Step 1: Find the amplitude and the number of cycles and period
|a| = | 2 | = 2 and b = 2, so it cycles 2 times from 0 to 2π. The period = 2π/b = 2π/2 = π.
Step 2: Divide the period into fourths. Identify Θ values for the five point pattern. π÷4 = π/4.  The x values (input) is 0, π/4, π/2, 3π/4 and π.
Step 3: Sketch the graph
The graph
Got it? #5 a, b
What is the graph of one cycle of each sine function?
A. y = 1.5 sin 2Θ
B. y = 3 sin π/2 Θ
Lesson check
Do you know HOW?
Handout #1,2
Do you UNDERSTAND?
#3, 4, 5
Exit slip
Sketch one cycle of the sine curve that has amplitude 2 and period π/2.

Friday, March 15, 2013

five pont 0-max-0-min-0 sine curve sketching


Topic: The Sine Function
Aim: How can we sketch the graph of a sine curve using 0-max-0-min-0 rule?
Do Now: Complete 1 – 9 on practice 13-4
Concept Summary
Suppose y = a sin , when a ≠ 0, b > 0, and Θ is in radians.
|a| is the amplitude of the function
B is the number of cycles in the interval from 0 to 2π
2π/b is the period of the function
Using zero-max-zero-min-zero five point pattern to sketch a sine curve
Look at Problem #4
Handout on page 846
Using y = a sin b Θ, where a = 2 and the period
Is 4π
We can sketch the sine curve using the five point summary as well as find the function rule
Got it? #4
After problem 4
Then look at Problem 5
What is the graph of one cycle of y = ½ sin 2Θ?
Exit slip
What is the amplitude and period of the function y = -3 sin ½ Θ?

Wednesday, March 13, 2013

cycles and periods of sine curves - 3/13/13


Topic: Periodic Functions

Aim: How can we find the period of a sine curve?
Do Now:
If Θ = π, find y, if y = sin Θ
The sine curve
 Find the sin π
Y = sin 4x
Finding the period of the sine curve
Use the graph of y = sin 4x to find:
A) how many cycles occur in the graph?
The graph shows 4 cycles
B) The period of the function:
  divide the interval of the graph by the number of cycles
2π /4 = π/2; therefore the period of y = sin 4x is π/2
Got it? Page 844 #2 a,b
How many cycles occur in the graph?
What is the period of the sine curve for each?
Exit slip
How do you find the number of cycles in the sine curve?
Identify the smallest repeating section of the graph and count the number of times it occurs.
For the graph below, find the number of cycles
The sine function
Y = sin Θ
Where Θ (angle in standard position) is matched with the y coordinate of a point on the unit circle.
This point is where the terminal side of the angle intersects the unit circle.
You can graph the sine function in radians or degrees (the x values or inputs)

Thursday, March 7, 2013

Finding the length of an intercepted arc - 3/7/13


Topic: Periodic Functions and Trigonometry

Aim: How can we find the length of an intercepted arc?

Do Now: Write 90° in radians. Then identify the cos and sin of the angle

Problem 3 on pg 838

Finding the length of an arc

Use the formula s = , where s = length of the arc, r = radius and θ = angle measure in radians

Example: if r = 3” and θ = 5π/6 radians, then s (the length of the arc) = (3)(5π/6) = /2. We now simplify to get estimated inches using calculator ≈ 7.9. Therefore the arc has a length of about 7.9”

Complete on your own

Got it? #3 a and b on handout

Then complete # 21 – 26 on back (Practice 13-3)

All problems on this sheet #1 – 26 will be collected today and count as 50% of an assessment; the other 50% will be an in class quiz on Monday!

Exit slip

How many radians are in  60°?•

How many degrees are in 5π/6 radians?



Sketch these angles in the unit circle and label their x and y coordinates.

Finding cosine and sine of a radian measure

Vocabulary

Central Angle: an angle of a circle with vertex at the center.

Intercepted arc: portion of the circle with endpoints on the sides of the central angle and remaining points within the interior of the angle

Radian: measure of a central angle that intercepts an arc with length equal to the radius of the circle.


Wednesday, March 6, 2013

Radian Measures 3/06


Topic: Periodic Functions and Trigonometry
Aim: How can we find exact values of cosine and sine using radian measures?
Do Now: What is the exact value of the cos π/4 radians?
Finding cos π/4 radians
π/4 radians = ¼ of a straight angle because ¼ x π = π/4. ¼ x 180 = 45.

π radians = 180° (straight angle)

Also, π/4 radians x 180/π radians = 45°

Cos π/4 = cos 45 = √2/2.
Finding cosine and sine of a radian measure
What are the exact values of cos (7π/6 radians) and sin (7π/6 radians)?
7π/6 radians x 180/π radians =

7 x 30 = 210°

Reference angle = 60° and in Quad III, where cos and sin are both negative (-)

Cos (7π/6 radians) = -  √3/2
Sin (7π/6 radians ) = - ½

Practice 13-3
Complete problems #1 – 20  ALL.
Exit slip
How many radians are in  60°?


How many degrees are in 5π/6 radians?


Sketch these angles in the unit circle and label their x and y coordinates.
Essential Understanding
An angle with a full circle rotation measures 2π radians.
An angle with a semicircle rotation measures π radians.

Recall: π = 3.14159……. (is this a rational or irrational number?)
Vocabulary (visualized)
Key Concept
Proportion relating radians and degrees

d°/180° = r radians/π radians

Use this proportion to convert between radians and degrees.
Why does the proportion work?
Recall that the circumference of a circle is 2πr, so there are 2π radians in any circle.

Since 2π radians = 360°, then π radians = 180°.

To convert degrees to radians, multiply by π radians/180°
To convert radians to degrees, multiply by 180°/π radians.
Vocabulary
Central Angle: an angle of a circle with vertex at the center.
Intercepted arc: portion of the circle with endpoints on the sides of the central angle and remaining points within the interior of the angle
Radian: measure of a central angle that intercepts an arc with length equal to the radius of the circle.
Using conversion factors/dimensional analysis
Problem 1:

What is the degree measure of an angle of -3π/4 radians?
Solution:
-3π/4 radians = -3π/4 radians X 180°/π radians

An angle of -3π/4 radians = -135°

Problem 2
What is the radian measure of an angle of 27°?

Solution:

27° = 27° X π radians/180° = 3π/20 radians.
An angle of 27° measures 3π/20 radians.
Complete the 4 questions on handout page 837 (#1 a-d)
a. π/2 radians = _____ °

b. 225° = ______ radians

c. 2 radians = _____ °

d. 150° = _____ radians