Sir I just wanted to say goodbye, I couldnt personally say it in school I was in a rush, but I wanted you to know this is one of the, if not the best year I had. I hadnt learned so much in a long time. I hope you have a great life and keep changing peoples life, Good Luck and thanks for everything.
Jaime
Thursday, June 21, 2007
Good Bye!
Dear Rumidog!
I know I talk in the name of everyone when I say you will be missed! Thank you very much for what you taught us this year, you were an excellent teacher, and I personally loved to go to your class. I wish you the best wherever you go in life, and don't forget to come to visit!
BYE!!!!
Larisa! (the "super" monitor)
I know I talk in the name of everyone when I say you will be missed! Thank you very much for what you taught us this year, you were an excellent teacher, and I personally loved to go to your class. I wish you the best wherever you go in life, and don't forget to come to visit!
BYE!!!!
Larisa! (the "super" monitor)
Tuesday, June 12, 2007
Little Mistake
On problem #5, csc is 30 degrees, not 25 degrees. (francisco sevilla and martha alcocer)
Series Question!!
Can a series diverge with a finite sum? Its that I am tring to find out when the absolute value of r is greater than 1, the series diverges, but I do not know when does it goes to infinity or has a finite sum. Can someone help me..Thanx
Final Assignment -
Final Assignment
By: Martha L. Alcocer and Francisco Sevilla
1. lim ((9x^2) + 3x)/(4x - (5x^3))
(as x approaches infinity)
lim
(as x approaches infinity)
((9x^2) + 3x)/(4x - (5x^3)) x (1/x^3)/(1/x^3) =
((9/x^0)+(3/x^2))/((4/x^2)+5) = 0.
2.
CSC 25° =
r/y = 1/(1/2) = 2.
3. Turn 75° to radians, and 1.2 radians to degrees.
75° x pie/180 = 1.31 radians.
1.2 x 180/pie = 68.76°
4. Find all 1st revolution angles for all angles[0,360]:
9sec o - 9 = 2
sec o = 11/4
(sec -1) sec o = 11/4 (sec -1)
o = sec -1 (11/4) = cos (4/11) = .99°
360° - 0.99° = 359.001°
5. Find the other five trig functions if cot 2/1 is on the 1st quadrant.
cot = 2/1 = x/y = 2 x^2+y^2 = r^2 = 4 + 1 = r^2 = squared root of 5.
tan = 1/2
sec = squared root of 5/2
cos = 2/squared root of 5
csc = squared root of 5/1
sin = 1/ squared root of 5
6. 10 - 3tanX = 8 + 2tanX
-3tanX = 8 + 2tanX - 10
-3tanX = 2tanX -2
-1tanX = -2
tanX = 2/1
(tan -1) x = 2tanX (tan -1)
x = 63.44
7. Mr. Alcantara told his students that they couldnt pass out of a certain sector of the trailer. If that sector has a radius of 5m and 1.2 radians. How long is the area of the sector in which the students can be?
s= (5)(1.2)
s= 6m
k= (1/2)(5)(6)
k= 15 m^2
8. Francisco is a very spoiled boy. His parents give him each christmas twice the presents they gave him the year before. His first year his parents gave him one present. Right now he is 17. How many presents has he received?
1,2,4,8,....(n=17)
s17= 1(1-(2^17))/(1-2)
s17=131,071 presents.
9. 3 x 4 + 4 x 5 + 5 x 6 + 6 x 7.... 30 x 31
tn= 3 + (n-1)(1) = 3 + n - 1 = 2 + n
tn= 4 + (n-1)(1) = 4 + n - 1 = n + 3
(n+2)(n+3)
n^2+3n+2n+ 6
n(n+1)(2n+1))/(6) + 5(n(n+1))/(2)
(28x29x57)/(6) + 5((28x29)/(2)) + 6.28
= 3654 + 2030 + 168 = 5852
10. 1/2 + 1/4 + 1/8 +....
Sn = t1/(1-r)
Sn = (1/2)/(1-0.5)
Sn = 1
By: Martha L. Alcocer and Francisco Sevilla
1. lim ((9x^2) + 3x)/(4x - (5x^3))
(as x approaches infinity)
lim
(as x approaches infinity)
((9x^2) + 3x)/(4x - (5x^3)) x (1/x^3)/(1/x^3) =
((9/x^0)+(3/x^2))/((4/x^2)+5) = 0.
2.
CSC 25° =
r/y = 1/(1/2) = 2.
3. Turn 75° to radians, and 1.2 radians to degrees.
75° x pie/180 = 1.31 radians.
1.2 x 180/pie = 68.76°
4. Find all 1st revolution angles for all angles[0,360]:
9sec o - 9 = 2
sec o = 11/4
(sec -1) sec o = 11/4 (sec -1)
o = sec -1 (11/4) = cos (4/11) = .99°
360° - 0.99° = 359.001°
5. Find the other five trig functions if cot 2/1 is on the 1st quadrant.
cot = 2/1 = x/y = 2 x^2+y^2 = r^2 = 4 + 1 = r^2 = squared root of 5.
tan = 1/2
sec = squared root of 5/2
cos = 2/squared root of 5
csc = squared root of 5/1
sin = 1/ squared root of 5
6. 10 - 3tanX = 8 + 2tanX
-3tanX = 8 + 2tanX - 10
-3tanX = 2tanX -2
-1tanX = -2
tanX = 2/1
(tan -1) x = 2tanX (tan -1)
x = 63.44
7. Mr. Alcantara told his students that they couldnt pass out of a certain sector of the trailer. If that sector has a radius of 5m and 1.2 radians. How long is the area of the sector in which the students can be?
s= (5)(1.2)
s= 6m
k= (1/2)(5)(6)
k= 15 m^2
8. Francisco is a very spoiled boy. His parents give him each christmas twice the presents they gave him the year before. His first year his parents gave him one present. Right now he is 17. How many presents has he received?
1,2,4,8,....(n=17)
s17= 1(1-(2^17))/(1-2)
s17=131,071 presents.
9. 3 x 4 + 4 x 5 + 5 x 6 + 6 x 7.... 30 x 31
tn= 3 + (n-1)(1) = 3 + n - 1 = 2 + n
tn= 4 + (n-1)(1) = 4 + n - 1 = n + 3
(n+2)(n+3)
n^2+3n+2n+ 6
n(n+1)(2n+1))/(6) + 5(n(n+1))/(2)
(28x29x57)/(6) + 5((28x29)/(2)) + 6.28
= 3654 + 2030 + 168 = 5852
10. 1/2 + 1/4 + 1/8 +....
Sn = t1/(1-r)
Sn = (1/2)/(1-0.5)
Sn = 1
Final Exam & Book Return
Hola, 11B.
Good luck tomorrow; it's easy.
Be sure to bring your textbook and your checkout sheet to the final. Otherwise you will have a hard time finding me. I will be leaving campus to grade your exams.
Remember, some of you must post your final assignment by 5:00 pm.
See you at 7:15
Good luck tomorrow; it's easy.
Be sure to bring your textbook and your checkout sheet to the final. Otherwise you will have a hard time finding me. I will be leaving campus to grade your exams.
Remember, some of you must post your final assignment by 5:00 pm.
See you at 7:15
Blog Final Problems
3.Simplify and find the sum.
10
∑ 2+(n-1)5
n=1
10
∑5n - ∑3
n=1
10
5∑n - ∑3
n=1
5(11)(5) - 30
275 - 30 = 45
5. Find sin of a 30 degree angle.
Y= 1, X= √3, R = 2
Sin= Y/r, so sin= 1/2
6. Convert 45 degrees in radians.
450 degrees * л/180 degrees = 2.5л or 5л/2
7. Find all first revolution angles if cot= 3/5
cot=x/y, tan=y/x
tan= 5/3, then you get its inverse.
so the angle= tan-1 (5/3)
angle= 59.03, 239.04
8.Find the recusrive and explcit definition for the sequence 2,4,6,8,10. Then find T20.
2,4,6,8,10
2 2 2 2
t1=2, d=2
Explicit= T1 + (n-1)d
= 2+ (n-1)d
= 2n
Recursive= T1=2
Tn=T(n-1) +2
T20 = 2(20)
t20 = 40
9. Find the sum of 2,1,0.5,0.25,0.125
1/2 = 0.5, 0.5/1 = 0/5
Rate = 0.5
Sn=t1/1-r, Sn= 2/0.5
Sum =4
10.Find 5 trig functions if cot=3/4 in the 1st quadrant.
Cot=x/y, so y=4 and x=3
Using pythagorean theorem, find r.
3^2+4^2=r^2
9+16=r^2
25=r^2
√25=r
5=r
sin=y/r or 4/5
cos=x/r or 3/5
tan=y/x or 4/3
sec=r/x or 5/3
csc=r/y or 5/4
difference= 2
t1=1
Explicit formula = 1 + (n-1)2
= 2n-1
8
∑ 2n-1
n=1
8
∑2n - ∑1
n=1
8
2∑n - 8
n=1
72-8 = 64
10
∑ 2+(n-1)5
n=1
10
∑5n - ∑3
n=1
10
5∑n - ∑3
n=1
5(11)(5) - 30
275 - 30 = 45
5. Find sin of a 30 degree angle.
Y= 1, X= √3, R = 2
Sin= Y/r, so sin= 1/2
6. Convert 45 degrees in radians.
450 degrees * л/180 degrees = 2.5л or 5л/2
7. Find all first revolution angles if cot= 3/5
cot=x/y, tan=y/x
tan= 5/3, then you get its inverse.
so the angle= tan-1 (5/3)
angle= 59.03, 239.04
8.Find the recusrive and explcit definition for the sequence 2,4,6,8,10. Then find T20.
2,4,6,8,10
2 2 2 2
t1=2, d=2
Explicit= T1 + (n-1)d
= 2+ (n-1)d
= 2n
Recursive= T1=2
Tn=T(n-1) +2
T20 = 2(20)
t20 = 40
9. Find the sum of 2,1,0.5,0.25,0.125
1/2 = 0.5, 0.5/1 = 0/5
Rate = 0.5
Sn=t1/1-r, Sn= 2/0.5
Sum =4
10.Find 5 trig functions if cot=3/4 in the 1st quadrant.
Cot=x/y, so y=4 and x=3
Using pythagorean theorem, find r.
3^2+4^2=r^2
9+16=r^2
25=r^2
√25=r
5=r
sin=y/r or 4/5
cos=x/r or 3/5
tan=y/x or 4/3
sec=r/x or 5/3
csc=r/y or 5/4
11. Solve for 2sin -9 =3
2sin = 12
sin = 6
Since sin = y/r, and y cant be greater than the radius, this is undefined.
12.A circle has a radius of 30cm. Find the sector of a 30 degree angle.
s=r * angle
convert the angle in radians
30 degree * π/180 degrees = π/6
π/6 * 30cm = 15.7 cm
difference= 2
t1=1
Explicit formula = 1 + (n-1)2
= 2n-1
8
∑ 2n-1
n=1
8
∑2n - ∑1
n=1
8
2∑n - 8
n=1
72-8 = 64
Monday, June 11, 2007
Assignment
Juliana Lecompte and Sofia Schuster
Assignment
1.
Lim 2x2 + x + 3 / 5x3 + x2 + 2x – 4
n to ∞
Lim 2x2 + x + 3 / 5x3 + x2 + 2x – 4 * 1/x3 /1/x3
n to ∞
Lim 2/x + 1/x2 + 3/x3/ 5 + 1/x + 2/x2 – 4/x3
n to ∞
Lim 0/5 = 0
n to ∞
2.
Finite Sum
10+5+2.5+1.25
R= .5
S∞ = T1/ 1-R S∞ = 10/.5 = 20
Not Finite Sum
1+2+4+8+16
R=2
S∞ = 1(1-2n/1-R) S∞= 1(1- ∞/-1) Turns positive so: S∞ = ∞ = DNE
3.
Simplify and Evaluate
6
∑ 4+ (n-1)4
n=1
6 6 6
∑ 4 + 4 ∑n - ∑1
n=1 n=1 n=1
24+ 4[3(5) – 6]
24+ 4 (15-6)
24+4(9) = 60
5.
Find the value of any trig function
√ 2
1
1
Sin 45 = 1/ √2 * √2/√2= √2/ 2
6.
Convert
150 to radians π
150 * π/ 180 = 2.68
5 radians to Degrees
5* 180/π = 286.5
7.
Find first rev angles
Cos Ө = 4/5
Cos -1= 36.90
360 – 36.9 = 323.10
Cos it’s positive in the 1st and 4th quadrants
8.
Recursive and Explicit definition
5, 9, 13, 17
D= 4
Recursive def = Start in 5 and add 4 to each term it gives you.
T1= 5
Tn = (Tn-1)+4
T5 = (T5-1) +4
T5 = T4 + 4
T5 = 17 + 4
T5= 21
Explicit def
Tn = 5 + (n-1)4
T6 = 5 + (6-1) 4
T6 =5 + 5(4)
T6 = 25
9.
Find the sum of finite and infinite geometric and arithmetic series.
Arithmetic Finite
1+2+3+4+…+20
S20 = 20/2 (1+20)
S20 = 10(21)
S20= 210
Geo infinite
20 + 5+1.25
S∞ = T1/ 1-R
S∞ = 20/.75= 26.7
10.
Tan Ө = -5/4 quadrant # 2
52 + 42 = R2
√41= R
R= 6.5
Cot Ө = 4/5
Cos Ө = -4/6.5
Sin Ө= 5/6.5
Csc Ө = -6.5/5
Sec Ө= 6.5/4
14.
8 + 10 + 12 +14 +…+22
n
∑8 +(n-1)2
n=1
(22-8/2)+1 = n
n= 8
8 8 8
∑8 + 2∑n - ∑1
n=1 n=1 n=1
64+2[4(9)-8]
64+ 56= 120
Assignment
1.
Lim 2x2 + x + 3 / 5x3 + x2 + 2x – 4
n to ∞
Lim 2x2 + x + 3 / 5x3 + x2 + 2x – 4 * 1/x3 /1/x3
n to ∞
Lim 2/x + 1/x2 + 3/x3/ 5 + 1/x + 2/x2 – 4/x3
n to ∞
Lim 0/5 = 0
n to ∞
2.
Finite Sum
10+5+2.5+1.25
R= .5
S∞ = T1/ 1-R S∞ = 10/.5 = 20
Not Finite Sum
1+2+4+8+16
R=2
S∞ = 1(1-2n/1-R) S∞= 1(1- ∞/-1) Turns positive so: S∞ = ∞ = DNE
3.
Simplify and Evaluate
6
∑ 4+ (n-1)4
n=1
6 6 6
∑ 4 + 4 ∑n - ∑1
n=1 n=1 n=1
24+ 4[3(5) – 6]
24+ 4 (15-6)
24+4(9) = 60
5.
Find the value of any trig function
√ 2
1
1
Sin 45 = 1/ √2 * √2/√2= √2/ 2
6.
Convert
150 to radians π
150 * π/ 180 = 2.68
5 radians to Degrees
5* 180/π = 286.5
7.
Find first rev angles
Cos Ө = 4/5
Cos -1= 36.90
360 – 36.9 = 323.10
Cos it’s positive in the 1st and 4th quadrants
8.
Recursive and Explicit definition
5, 9, 13, 17
D= 4
Recursive def = Start in 5 and add 4 to each term it gives you.
T1= 5
Tn = (Tn-1)+4
T5 = (T5-1) +4
T5 = T4 + 4
T5 = 17 + 4
T5= 21
Explicit def
Tn = 5 + (n-1)4
T6 = 5 + (6-1) 4
T6 =5 + 5(4)
T6 = 25
9.
Find the sum of finite and infinite geometric and arithmetic series.
Arithmetic Finite
1+2+3+4+…+20
S20 = 20/2 (1+20)
S20 = 10(21)
S20= 210
Geo infinite
20 + 5+1.25
S∞ = T1/ 1-R
S∞ = 20/.75= 26.7
10.
Tan Ө = -5/4 quadrant # 2
52 + 42 = R2
√41= R
R= 6.5
Cot Ө = 4/5
Cos Ө = -4/6.5
Sin Ө= 5/6.5
Csc Ө = -6.5/5
Sec Ө= 6.5/4
14.
8 + 10 + 12 +14 +…+22
n
∑8 +(n-1)2
n=1
(22-8/2)+1 = n
n= 8
8 8 8
∑8 + 2∑n - ∑1
n=1 n=1 n=1
64+2[4(9)-8]
64+ 56= 120
Final Assignment
By: Larisa Jasbon and Kristina Wick
In order to see the larger image, click on the BubbleShare. If anything is not clear, please, let us know.
Problems:
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Answers:
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We hope that these problems are a helpful study tool for the final exam. If you have any questions, we are willing to help you in any way that we can.
Sayonara!!!
In order to see the larger image, click on the BubbleShare. If anything is not clear, please, let us know.
Problems:
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Answers:
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We hope that these problems are a helpful study tool for the final exam. If you have any questions, we are willing to help you in any way that we can.
Sayonara!!!
Radians and Degrees
and 
Radian measure!!
Radian Measure
Converting from radians to degrees
IN THE RADIAN SYSTEM of angular measurement, the measure of one revolution is 2π.
Half a circle, then, is π. And, most important, each right angle is half of π: π/2.
Three right angles will be: 3· π/2= 3π/2.
Five right angles will be 5π/2. And so on.
Radians into Degrees
Students should have a clear picture of this:
π/4 is half of π/2, so it equals a 45°.
π/4 is one fourth of π.
π/3 is one third. Which equals 180° ÷ 3 = 60°.
π/6 is one sixth: 180° ÷ 6 = 30°.
5π/4
= 5 · 180/4
= 5· 45°
= 225°.
2π/3 is a third of 2π. So 360° ÷ 3 = 120°.
***To convert from radians to degrees, you convert the pie π into degrees : 180° and solve for the angle.
If it is π/4 angle, you divide 180 into 4 which is 45. Therefore π divided 4 times equals a 45°.
Solve:
Convert from radian measures into degrees:
a) π/8
b) 2π/5
c) 7π/4
d)9π/2
e)4π/3
f)5π/6
g)7π/9
ANSWERS IN COMMENTS!!
Converting from radians to degrees
IN THE RADIAN SYSTEM of angular measurement, the measure of one revolution is 2π.
Half a circle, then, is π. And, most important, each right angle is half of π: π/2.
Three right angles will be: 3· π/2= 3π/2.
Five right angles will be 5π/2. And so on.
Radians into Degrees
Students should have a clear picture of this:
π/4 is half of π/2, so it equals a 45°.
π/4 is one fourth of π.
π/3 is one third. Which equals 180° ÷ 3 = 60°.
π/6 is one sixth: 180° ÷ 6 = 30°.
5π/4
= 5 · 180/4
= 5· 45°
= 225°.
2π/3 is a third of 2π. So 360° ÷ 3 = 120°.
***To convert from radians to degrees, you convert the pie π into degrees : 180° and solve for the angle.
If it is π/4 angle, you divide 180 into 4 which is 45. Therefore π divided 4 times equals a 45°.
Solve:
Convert from radian measures into degrees:
a) π/8
b) 2π/5
c) 7π/4
d)9π/2
e)4π/3
f)5π/6
g)7π/9
ANSWERS IN COMMENTS!!
Arc length
>Practice Problems
a) At a central angle of 2.35 radians, what ratio has the arc to the radius?
b) In which quadrant of the circle does 2.35 radians fall?
c) If the radius is 10 cm, and the central angle is 2.35 radians, then how long is the arc?
*****ANSWERS ARE IN COMMENTS!!**********
Problem 1
a) At a central angle of π /5, approximately what ratio has the arc to the radius? Take π=3.
b) If the radius is 15 cm, approximately how long is the arc?
Problem 2
In a circle whose radius is 4 cm, find the arc length intercepted by each of these angles. Again, take π = 3.
a) At a central angle of 2.35 radians, what ratio has the arc to the radius?
b) In which quadrant of the circle does 2.35 radians fall?
c) If the radius is 10 cm, and the central angle is 2.35 radians, then how long is the arc?
*****ANSWERS ARE IN COMMENTS!!**********
Problem 1
a) At a central angle of π /5, approximately what ratio has the arc to the radius? Take π=3.
b) If the radius is 15 cm, approximately how long is the arc?
Problem 2
In a circle whose radius is 4 cm, find the arc length intercepted by each of these angles. Again, take π = 3.
Trigonometry- Arc Length
The definition of a radian meadure
An angle of a 1 radian
Examples!!
Let the letter s (for space) symbolize the length of an arc, which is called arc length.
Now the circumference of a circle is an arc length.
The ratio of the circumference to the diameter is the basis of radian measure.
That ratio is the definition of π.
π = C/D
C = Circumference
D = Diameter
Since D = 2r, then
π = C/2r
or,
2π = C/r
That ratio of the circumference of a circle C to the radius r -- 2π -- is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.
Radian measure = θ s /r
Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that indicates the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle θ.
EXAMPLE:
In a circle whose radius is 10 cm, a central angle θ intercepts an arc of 8 cm.
a)What is the radian measure of that angle?
b) what is the arc length if the radius is 5 cm?
Answers
a)
θ = s /r
= 8/ 10
= .8
b)For a given central angle, the ratio of arc to radius is the same. 5 is half of 10. Therefore the arc length will be half of 8: 4cm.
An angle of 1 radian
Note that an angle of 1 radian is a central angle whose subtending arc is equal in length to the radius.
An angle of a 1 radian
Examples!!
Let the letter s (for space) symbolize the length of an arc, which is called arc length.
Now the circumference of a circle is an arc length.
The ratio of the circumference to the diameter is the basis of radian measure.
That ratio is the definition of π.
π = C/D
C = Circumference
D = Diameter
Since D = 2r, then
π = C/2r
or,
2π = C/r
That ratio of the circumference of a circle C to the radius r -- 2π -- is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.
Radian measure = θ s /r
Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that indicates the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle θ.
EXAMPLE:
In a circle whose radius is 10 cm, a central angle θ intercepts an arc of 8 cm.
a)What is the radian measure of that angle?
b) what is the arc length if the radius is 5 cm?
Answers
a)
θ = s /r
= 8/ 10
= .8
b)For a given central angle, the ratio of arc to radius is the same. 5 is half of 10. Therefore the arc length will be half of 8: 4cm.
An angle of 1 radian
Note that an angle of 1 radian is a central angle whose subtending arc is equal in length to the radius.
Sunday, June 10, 2007
TRIG functions!!
Cosine Overview
A type of trig. function
Definition 1 is the simplest and most intuitive definition of the cosine function. It basically says that, on a right triangle, the following measurements are related:
the length of the triangle's hypotenuse
the length of one of the other sides
the measurement of the angle (q) adjacent to that other side
Graph of Cosine and Sine
Fundamental Trigonometric Identity
sin^2t + cos^2t = 1
A type of trig. function
Definition 1 is the simplest and most intuitive definition of the cosine function. It basically says that, on a right triangle, the following measurements are related:
the length of the triangle's hypotenuse
the length of one of the other sides
the measurement of the angle (q) adjacent to that other side
Graph of Cosine and Sine
Fundamental Trigonometric Identity
sin^2t + cos^2t = 1
Saturday, June 9, 2007
Study ..Final Exam
http://en.wikibooks.org/wiki/Trigonometry/Solving_Trigonometric_Equations
http://www.analyzemath.com/Trigonometry.html
http://www.analyzemath.com/Trigonometry.html
Thursday, June 7, 2007
Final Topic List & Assignment
Below you will find a list of skills that you will need for the final exam.
For those of you who did not write two scribe posts, you have the following assignment:
For those of you who did not write two scribe posts, you have the following assignment:
- Choose 10 of the 14 topics
- Write an example problem for each type of problem in the list.
- Number the problems with the numbers of the problem types found on the study list below.
- Solve your problems.
- Turn in a hard copy of your problems and solutions at the beginning of class on Tuesday.
- Post your problems and solutions to the blog by 5:00 pm Tuesday ( I will check at 5:00)
You can do this assignment with a partner or by yourself.
Look back in your notes, text, and quizzes for ideas. Be sure that you create your own problems. Each group must have different problems. Suspected copying from the Internet, from other students, or elsewhere, will be rewarded with a zero. Also, such an approach would not be good preparation for your final exam.
Remember, this assignment is designed to help you and the rest of your class. If you do a good job on this, then the final should go quite smoothly for all.
Wednesday, June 6, 2007
Trig. Graphs!!
Study for Trig. Graphs:
Sec Graph!!
A graph of sec(x). sec(x) is defined as 1/cos (x)
Cos Graph!!
A graph of csc(x). csc(x) is defined as 1/ sin(x)
Sine Graph!!
Careful analysis of this graph will show that the graph corresponds to the unit circle. X is essentially the degree measure(in radians), while Y is the value of the sine function.
Cos Graph!!
As with the sine function, analysis of the cosine function will show that the graph corresponds to the unit circle. One of the most important differences between the sine and cosine functions is that sine is an odd function while cosine is an even function.
Sine and cosine are periodic functions; that is, the above is repeated for preceding and following intervals with length 2π.
Cot Graph!!
A graph of cot(x). cot(x) is defined as 1/tan (x) or cos(x)/sin(x)
Tan graph!!
A graph of tan(x). tan(x) is defined as sin (x)/cos (x)
Sec Graph!!
A graph of sec(x). sec(x) is defined as 1/cos (x)
Cos Graph!!
A graph of csc(x). csc(x) is defined as 1/ sin(x)
Sine Graph!!
Careful analysis of this graph will show that the graph corresponds to the unit circle. X is essentially the degree measure(in radians), while Y is the value of the sine function.
Cos Graph!!
As with the sine function, analysis of the cosine function will show that the graph corresponds to the unit circle. One of the most important differences between the sine and cosine functions is that sine is an odd function while cosine is an even function.
Sine and cosine are periodic functions; that is, the above is repeated for preceding and following intervals with length 2π.
Cot Graph!!
A graph of cot(x). cot(x) is defined as 1/tan (x) or cos(x)/sin(x)
Tan graph!!
A graph of tan(x). tan(x) is defined as sin (x)/cos (x)
Review
STUDY FOR FINALS!!
Graph of Cubic function, where y=x^3
Polynomial Functions
Basic definitions:
1. When numbers are added or subtracted, they are called terms.
When numbers are multiplied, they are called factors.
2. A variable is a symbol that takes on values. A value is a number.
3. A constant is a symbol that has a single value.
4. A monomial in x is a single term of the form axn, where a is a real number and n is a whole number.
The whole numbers, recall, are the non-negative integers: 0, 1, 2, 3, 4, etc.
5. A polynomial in x is a sum of monomials in x.
6. The degree of a term is the sum of the exponents of all the variables in that term.
7. The leading term of a polynomial is the term of highest degree.
8. The leading coefficient of a polynomial is the coefficient of the leading term.
9. The degree of a polynomial is the degree of the leading term.
10.. The constant term of a polynomial is the term of degree 0; it is the term in which the variable does not appear.
Graph of Cubic function, where y=x^3
Polynomial FunctionsBasic definitions:
1. When numbers are added or subtracted, they are called terms.
When numbers are multiplied, they are called factors.
2. A variable is a symbol that takes on values. A value is a number.
3. A constant is a symbol that has a single value.
4. A monomial in x is a single term of the form axn, where a is a real number and n is a whole number.
The whole numbers, recall, are the non-negative integers: 0, 1, 2, 3, 4, etc.
5. A polynomial in x is a sum of monomials in x.
6. The degree of a term is the sum of the exponents of all the variables in that term.
7. The leading term of a polynomial is the term of highest degree.
8. The leading coefficient of a polynomial is the coefficient of the leading term.
9. The degree of a polynomial is the degree of the leading term.
10.. The constant term of a polynomial is the term of degree 0; it is the term in which the variable does not appear.
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