MA 154

NOTE: Some of the links below will be handouts that can be viewed and printed with your web browser. Others will be PDF documents that must be read with Adobe™ Acrobat™ Reader software. This software is free and available from Adobe's Web Site. 

Monday, January 12: Welcome to MA 154!

Introduction to the course and discussion of course policies. 
In-Class Activity: Oh Deer! 

For Wednesday, January 14
Read: Section 6.1 and course guide (and 3.1/3.2 Review)
Review: Section 3.1 -- 1, 3, 15, 167, 19, 25, 27 and 3.2 -- 5, 15, 17, 37

Wednesday, January 14: Periodic Functions
Graphed the height of a Ferris Wheel car as a function of time, reported midline, amplitude, period of what is called sinusoidal function. 
For Friday, January 16
Read: Section 6.2
Do: Section 6.1 -- 1-25 odd, 26, 27

Friday, January 16: Sine and Cosine
Defined the sine and cosine of an angle.
For Wednesday, January 21
Read: Section 6.3
Do: Section 6.2 -- 1, 5, 9, 11-27, 31-33

Wednesday, January 21: Radians  

Defined radian measurement. Suppose the vertex of an angle theta is at the center of a circle of radius r, and the angle spans an arc of length s.  The measure of the angle theta in radians is defined as the ratio of the arc length, s, to the radius r (See figure). It follows that an angle of 360 degrees spans an arc length of the circumference of the circle, s = 2πr, so this angle has a radian measure of = s / r = 2πr / r = 2π. Consequently 180 degrees is equivalent to π radians, which is a handy conversion factor. We looked at converting angles from degrees to radians and vice versa, with and without a calculator. Finally, we looked at some applied questions where we were asked to find the arc length if given the angle (in degrees) and the radius. It is paramount to convert theta from degrees to radians first before applying the relationship s = r*theta, since this only holds if theta is in radians.
For Friday, January 23 Read: Chapter 6 Tools page 288-291 and Section 6.4   Do: Section 6.3 -- 1-27 odd, 28-34, 36, 38, 39, 41. Also prepare for QUIZ 1 over 3.1/3.2 and 6.1, 6.2

Friday, January 23: Exact Values and Finding Angles
Took Quiz 1. Looked at exact values (see activity) of trig functions of 30º, 45º, 60º and their multiples, did some problems from the worksheet on finding angles in radians (exactly), and looked at right triangle definitions and applications. 
For Monday, January 26
Read: Section 6.4 and 6.5
Do:  Worksheet on finding the angle (Here's the key.) and Chapter 6 Tools page 288-291: 1-19, 21, 23, 29

Monday, January 26: Exact Values and Finding Angles cont'd
For Wednesday, January 28
Read: Section 6.4 and 6.5 and review Sections 5.1 - 5.3
Do: Prepare for QUIZ 2 over Section 6.3, Chapter 6 Tools, and Worksheet on Angles
No class meeting on Friday, January 30
HW 1 e-grade on Selected topics from Ch 6 due 11:59 pm Monday, February 2. 
You have unlimited attempts (until the due date) to obtain a perfect score.
Test 1 now is on Monday, February 9

Wednesday, January 28: Section 6.4
We discussed the domain, range, period, and amplitude of y = sin(x) and y = cos(x) and how these can be determined from the unit circle. We looked at an outside change to the function, which results in the original function being transformed vertically (change to the output).

1. y = Asin(x) and y = Acos(x) have amplitude |A|. 
   For A > 0, 
   the graph of y = Asin(x) vertically stretches or compresses the graph of  y = sin(x) by A units.
   the graph of y = –Asin(x) is a vertical reflection of the graph of y = Asin(x).
   Similarly for y = cos(x).  
   (See Section 5.2 and 5.3 for a review of these topics.)

2. y = sin(x) + k and y = cos(x) + k have midline k.
   For k > 0, 
   the graph of y = sin(x) + k vertically shifts the graph of y = sin(x) up k units.
   the graph of y = sin(x) – k vertically shifts the graph of y = sin(x) down k units.
   Similarly for y = cos(x).  
   (See Section 5.1 for a review of these topics.)

The first multiplies the output by a quantity; the second adds/subtracts a quantity to the output. In the next class we will look at the effect of doing both of these kind of transformations to the inside, which causes a horizontal transformation (change to the input).

Due for Friday, September 12
Read: Section 5.4 (to look at horizontal stretching) and 6.5 
Do: Section 6.4 -- 1-26 
No class meeting on Friday, January 30
HW 1 e-grade on Selected topics from Ch 6 due 11:59 pm Monday, February 2. 
You have unlimited attempts (until the due date) to obtain a perfect score.
Test 1 rescheduled on Monday, February 9. 

Monday, February 2: Section 6.5
One equation says it all: y = Asin(B(x - h)) + k
We explored the effects of  B and h on the graph.
For positive values of B and h: 
the period is 2π/B, the horizontal shift is h units to the right, and the phase shift is Bh. 
In other words, the cycle of the sine curve starts when x = h and ends when x = h + period. 
Due for Wednesday, February 4:
Read: Section 6.6 and 6.7 
Do: Section 6.5 -- 1- 37 student's choice
HW 1 e-grade on Selected topics from Ch 6 due 11:59 pm tonight Monday, Feb 2. 
You have unlimited attempts (until the due date) to obtain a perfect score.

Wednesday, February 4: Section 6.6
We investigated the tangent function and its properties, as well as defined the reciprocal functions csc x, sec x, and cot x.
Read:  Section 6.7
Do:  Section 6.6 -- 1-15 odd and be prepared for QUIZ 3 on 6.4 and 6.5: 
For Test 1 on Monday, Feb 9: Review old homework, quizzes and eGrade.  Here are some suggestions:
Section 3.1 -- 1, 3, 15, 167, 19, 25, 27
Section 3.2 -- 5, 15, 17, 37
Section 6.1: 13, 27
Section 6.2: 9, 27, 31
Section 6.3: 1-15 odd, 33, 37
Section 6.4: 11, 13. 17, 23
Section 6.5: 19, 25
Section 6.6: 1-15 odd
Chapter 6 Tools: 1-21 odd, 29
Chapter 6 Review: 1-30, 49-51
Check Your Understanding: page 285 1-64

Friday, February 6
Took QUIZ 3 and went over the KEY. Reviewed for Test 1 over 3.1, 3.2, 6.1-6.6, Ch 6 Tools on Monday, Feb. 9

Monday, February 9: Took Test 1

Wednesday, February 11: 
Worked on an activity involving the tangent function 
Read: Section 6.6 and 6.7 
Do: Parade Worksheet

Friday, February 13: 
A day spent with Pythagoras and SOHCAHTOA - what a nice pair!
Read:  Section 6.7 again and Section 7.1
Do:  Section 6.6 --  18-29 and Section 6.7 -- 1a, 2a, 18-22

Monday, February 16
Examined and used the inverse trig functions to find angles in degrees and radians.
Read:  Section 7.1
Do:  Section 6.7 -- 1-27, 33, 35-49 and Chapter 6 Review 34-43 odd

Wednesday, February 18
A brief jump to Section 7.5
Read:  Section 7.1 yet again
Do:  Section 7.5 -- 1-28

cos²x + 4sin x = 4
1 – sin²x + 4sin x = 4

The rest is algebra.... 

1 – sin²x + 4sin x = 4
–sin²x + 4sin x = 3
–sin²x + 4sin x – 3 = 0
sin²x – 4sin x + 3 = 0
(sin x – 3)((sin x – 1) = 0
sin x = 3 and sin x = 1
sin x = 3 has no solutions and sin x = 1 has a solution of π/2.

How nice! After introducing the double angle formulas, similar fun was done for solving cos 2x = cos x on  [0, 2π).
Replace cos 2x with 2cos²x – 1 and turn the crank....

cos 2x = cos x
2cos²x – 1 = cos x
2cos²x –  cos x – 1 = 0
(2cos x +  1)(cos x – 1) = 0
cos x = –1/2 and cos x = 1
x = 2π/3, 4π/3 and x = 0

Note that 2π is a solution but not in the requested interval of  [0, 2π).
Check the solution by graphing y = cos 2x  – cos x and watching it touch the x-axis at the three values 0, 2π/3, 4π/3. It was handy to set Xscl = π/3. You could also have graphed y = cos 2x and y = cos x and see when they intersect. Finally we looked at  Sum and Difference Identities handout.
Read: Section 8.1 
Do: Section 7.2 -- 28 and Sum and Difference Identities handout.
Want more practice solving quadratic equations? See Chapter 2 Tools pages 98ff 28-80, 87, 90, 94-96.