A. Sine and Cosine are similar in many ways. To get cosine, all you do is take the opposite of sine. It is proven through the trig chart.
Sin 0 = 0 Cos 0= 1
Sin pi/6 = ½ Cos pi/6=√3/2
Sin pi/4=√2/2 Cos pi/4=√2/2
Sin pi/3=√3/2 Cos pi/3= ½
Sin pi/2=1 Cos pi/2= 0
B. Tangent and Cotangent are also very similar. To get tangent, it is sin/cos, and to get cotangent all you have to do is flip tangent. This is also proven in the trig chart.
Tan 0 = 0 Cot 0 = undefined
Tan pi/6 = √3/3 Cot pi/6 = √3
Tan pi/4 = 1 Cot pi/4 = 1
Tan pi/3 = √3 Cot pi/3 = √3/3
Tan pi/2 = undefined Cot pi/2 = 0
C. Sine and Cosecant are partially different. To get Cosecant, just take the reciprocal of sine.
Sin 0 = 0 Csc 0 = undefined
Sin pi/6 = ½ Csc pi/6 = 2
Sin pi/4=√2/2 Csc pi/4 = √2
Sin pi/3=√3/2 Csc pi/3 = 2√3/3
Sin pi/2=1 Csc pi/2 = 1
D. Secant and Cosine are the same as Sine and Cosecant. In order to get Secant, just take the reciprocal of Cosine.
Cos 0= 1 Sec 0 = 1
Cos pi/6=√3/2 Sec pi/6 = 2√3/2
Cos pi/4=√2/2 Sec pi/4 = √2
Cos pi/3= ½ Sec pi/3 = 2
Cos pi/2= 0 Sec pi/2 = undefined
E. Sine, Cosine and Tangent are all alike because you can get each on of the angles from sine. All you have to remember is that to find cosine, you do the opposite of sine and to do tangent all you do is divide sine by cosine.
Tuesday, August 31, 2010
Nathan's Blog #2
A.) The relationship between sine and cosine is very easy. All you have to do is take the opposite of sine and you have cosine.
sin 0=0 - cos 0=1
sin pi/6=1/2 - cos pi/6=square root of 3/2
sin pi/4= square root of 2/2 - same for cos
sin pi/3=square root of 3/2 - cos pi/3=1/2
sin pi/2=1 - cos pi/2=0
B.) The next is tangent and cotangent. Tangent is the result of sin/cos. To get cotangent, just flip tangent.
tan 0=0 - cot 0=undefined
tan pi/6=square root of 3/3 - cot pi/6=square root of 3
tan pi/4=1 - same for cot
tan pi/3=square root of 3 - cot pi/3=square root of 3/3
tan pi/2=undefined - cot pi/2=0
C.) Our next pair is sine and cosecant. To find cosecant, just flip sine.
sin 0=0 - csc 0=undefined
sin pi/6=1/2 - csc pi/6=2
sin pi/4=square root of 2/2 - csc pi/4=square root of 2
sin pi/3=square root of 3/2 - csc pi/3=2 square root of 3/3
sin pi/2=1 - same for csc
D.) Cosine and secant are alike because all you have to do to find secant, is flip cosine.
cos 0=1 - same for sec
cos pi/6=square root of 3/2 - sec pi/6=2 square root of 3/2
cos pi/4=square root of 2/2 - sec pi/4=square root of 2
cos pi/3=1/2 - sec pi/3=2
cos pi/2=0 - sec pi/2=undefined
E.) The last relationship of the six trigonomic funtions is between sine, cosine, and tangent. To find cosine, it is the opposite of sine, and to find tangent, divide sine by cosine.
sin 0=0 - cos 0=1 - tan 0=0
sin pi/6=1/2 - cos pi/6=square root of 3/2 - tan pi/6=square root of 3/3
sin pi/4=square root of 2/2 - same for cosine - tan pi/4=1
sin pi/3=square root of 3/2 - cos pi/3=1/2 - tan pi/3=square root of 3
sin pi/2=1 - cos pi/2=0 - tan pi/2= undefined
sin 0=0 - cos 0=1
sin pi/6=1/2 - cos pi/6=square root of 3/2
sin pi/4= square root of 2/2 - same for cos
sin pi/3=square root of 3/2 - cos pi/3=1/2
sin pi/2=1 - cos pi/2=0
B.) The next is tangent and cotangent. Tangent is the result of sin/cos. To get cotangent, just flip tangent.
tan 0=0 - cot 0=undefined
tan pi/6=square root of 3/3 - cot pi/6=square root of 3
tan pi/4=1 - same for cot
tan pi/3=square root of 3 - cot pi/3=square root of 3/3
tan pi/2=undefined - cot pi/2=0
C.) Our next pair is sine and cosecant. To find cosecant, just flip sine.
sin 0=0 - csc 0=undefined
sin pi/6=1/2 - csc pi/6=2
sin pi/4=square root of 2/2 - csc pi/4=square root of 2
sin pi/3=square root of 3/2 - csc pi/3=2 square root of 3/3
sin pi/2=1 - same for csc
D.) Cosine and secant are alike because all you have to do to find secant, is flip cosine.
cos 0=1 - same for sec
cos pi/6=square root of 3/2 - sec pi/6=2 square root of 3/2
cos pi/4=square root of 2/2 - sec pi/4=square root of 2
cos pi/3=1/2 - sec pi/3=2
cos pi/2=0 - sec pi/2=undefined
E.) The last relationship of the six trigonomic funtions is between sine, cosine, and tangent. To find cosine, it is the opposite of sine, and to find tangent, divide sine by cosine.
sin 0=0 - cos 0=1 - tan 0=0
sin pi/6=1/2 - cos pi/6=square root of 3/2 - tan pi/6=square root of 3/3
sin pi/4=square root of 2/2 - same for cosine - tan pi/4=1
sin pi/3=square root of 3/2 - cos pi/3=1/2 - tan pi/3=square root of 3
sin pi/2=1 - cos pi/2=0 - tan pi/2= undefined
Charlie's Blog #2
A.) Cosine and sine have a relationship in which, using the Pythagorean Theorem, sinΘ + cosΘ = 1. In any angle using the theorem they will always equal 1. [http://www.math.dartmouth.edu/opencalc2/cole/lecture9.pdf]
Also, on the trigonometry chart, sine is the opposite order of cosine. All the numbers and factions are the same, just opposite.
*Sin 0 = 0 Cos 0 = 1
Sin pi/6 = ½ Cos pi/6 = square root of 3/2
Sin pi/4 = square root of 2/2 Cos pi/4 = square root of 2/2
Sin pi/3 = square root of 3/2 Cos pi/3 = 1/2
Sin pi/2 = 1 Cos pi/2 = 0
B.) Tangent and Cotangent are in relation with there values. Where tan = y/x and cot = x/y. So if tan equals 4/9 then cot equals 9/4.
There is also the relationship on the trigonometry chart. Like sine and cosine, these are also opposite of each other.
*Tan 0 = 0 Cot 0 = 0
Tan pi/6 = square root of 3/3 Cot pi/6 = square root of 3
Tan pi/4 = 1 Cot pi/4 = 1
Tan pi/3 = square root of 3 Cot pi/3 = square root of 3/3
Tan pi/2 = undefined Cot pi/2 = 0
C.) Sine and Cosecant relate in the way that sin = y/r and csc = r/y.
So on the ‘Trig Chart’, sine flipped equals cosecant.
*Csc 0 = undefined -instead of 0
Csc pi/6 = 2 -instead of ½
Csc pi/4 = square root of 2 -instead of square root of 2/2
Csc pi/3 = 2 square root of 3/2 -instead of square root of 3/2
Csc pi/2 = 1 -the same as sine
D.) Secant and Cosine are related in the way that cos = x/r while sec = r/x.
And on the trigonometry chart, like sine and cosecant, it’s cosine flipped.
*Sec 0 = 1 -the same as cosine
Sec pi/6 = 2 square root of 3/3 -instead of square root of 3/2
Sec pi/4 = square root of 2 -instead of square root of 2/2
Sec pi/3 = 2 -instead of ½
Sec pi/2 = undefined -instead of 0
D.) Sine, Cosine, and tangent are alike because tangent can be found if given sine and/or cosine. Tangent equals y/x, sine equals y/r, and cosine equals x/r. Therefore if sinΘ = 3/6 and cosΘ = 9/6, then tan = 3/9.
Also, on the trigonometry chart, sine is the opposite order of cosine. All the numbers and factions are the same, just opposite.
*Sin 0 = 0 Cos 0 = 1
Sin pi/6 = ½ Cos pi/6 = square root of 3/2
Sin pi/4 = square root of 2/2 Cos pi/4 = square root of 2/2
Sin pi/3 = square root of 3/2 Cos pi/3 = 1/2
Sin pi/2 = 1 Cos pi/2 = 0
B.) Tangent and Cotangent are in relation with there values. Where tan = y/x and cot = x/y. So if tan equals 4/9 then cot equals 9/4.
There is also the relationship on the trigonometry chart. Like sine and cosine, these are also opposite of each other.
*Tan 0 = 0 Cot 0 = 0
Tan pi/6 = square root of 3/3 Cot pi/6 = square root of 3
Tan pi/4 = 1 Cot pi/4 = 1
Tan pi/3 = square root of 3 Cot pi/3 = square root of 3/3
Tan pi/2 = undefined Cot pi/2 = 0
C.) Sine and Cosecant relate in the way that sin = y/r and csc = r/y.
So on the ‘Trig Chart’, sine flipped equals cosecant.
*Csc 0 = undefined -instead of 0
Csc pi/6 = 2 -instead of ½
Csc pi/4 = square root of 2 -instead of square root of 2/2
Csc pi/3 = 2 square root of 3/2 -instead of square root of 3/2
Csc pi/2 = 1 -the same as sine
D.) Secant and Cosine are related in the way that cos = x/r while sec = r/x.
And on the trigonometry chart, like sine and cosecant, it’s cosine flipped.
*Sec 0 = 1 -the same as cosine
Sec pi/6 = 2 square root of 3/3 -instead of square root of 3/2
Sec pi/4 = square root of 2 -instead of square root of 2/2
Sec pi/3 = 2 -instead of ½
Sec pi/2 = undefined -instead of 0
D.) Sine, Cosine, and tangent are alike because tangent can be found if given sine and/or cosine. Tangent equals y/x, sine equals y/r, and cosine equals x/r. Therefore if sinΘ = 3/6 and cosΘ = 9/6, then tan = 3/9.
Nicala's blog
ok one of the things we learned in class this week was reference angles & it was one of the easiest thing we did thats why i picked it. :-)
step 1. figure out what quadrant sin or cos is in. if you don't know what quadrant its in you won't know if the answer is negative or postive and that will throw off the answer.
step 2. subtract by 180 until you get yr answer between 0 & 90 & if you get a negative # use absolutely value to make the # postive.
{hint:if yr answer is already between 0 & 90 then you don't need to subtract.}
step 3. if yr answer is 30,45,60,or 90. then you need to use yr trig chart to figure out the exact of yr answer.
Example: Sin 225
step 1. figure out what quadrant sin 225. its in quadrant 3 which makes it negative.
{another hint: quadrant 1 0- 90 quadrant 2 91-180 quadrant 181- 270 quadrant 271- 360}
step 2. subtract by 180. 225- 180=45
step 3. next you need to find the exact value 45 on yr trig chart. 45= pi/4 on yr trig chart.
Monday, August 30, 2010
Week 1 Blog Prompt
Explain the relationship between the following trig functions:
a. Sine and Cosine
b. Tangent and Cotangent
c. Sine and Cosecant
d. Cosine and Secant
e. Sine, Cosine & Tangent
a. Sine and Cosine
b. Tangent and Cotangent
c. Sine and Cosecant
d. Cosine and Secant
e. Sine, Cosine & Tangent
Sunday, August 29, 2010
BLAAAAHHHHH!!!
This is David on Mary's blog thing. I don't feel like creating my own account so I'm just going to share this one with Mary.
But actually having something to do with the class, I need to take better notes. I am having a bit of trouble with 7-1 through, well, everything but 7-4.
I get most of the sine and cosine functions. I understand the unit circle and trig chart, for the most part, but I don't have the formulas down for the word problems and whatnot that was shown to us in class.
Getting back on track, lets talk about something simple, like converting degrees to radians and vice versa!
This is clearly one of the more simple, if not the simplest, thing we will be doing in this class all year. :'(
No one else should have trouble with this if I can understand it.
All you need to know are two easy little formulas. I'll start with Degrees to Radians, give an example and then do the same with Radians to Degrees.
Degrees to Radians = pi/180
Ex. Convert 72 degrees to radians.
Instead of plugging it into your calculator just put the function down on paper and solve by treating pi as a variable.
So you would proceed with 72/1 x pi/180
Multiply straight across 72 x pi
1 x 180
And your product comes out to be 72 pi/180
That reduces to 2 pi/5
And that is your answer, if you can't reduce in your head just plug the fraction into the calculator without the pi in the function.
On the Flip side!
If you need to convert Radians to Degrees you just use the reciprocal of the first formula.
Radians to Degrees = 180/ pi
Ex. Convert 4 pi/ 2 to degrees
You set up the equation 4 pi x 180
2 x pi
Now you can clearly see that you can cancel out the bottom half of the equation.
The pi's cancel out and since 180 is divisible by 2 it can be reduced.
You are then left with 4 x 90
Your answer is then 360 Degrees
That is it for tonight's lesson, I sincerely hope that it helped someone in some way.
But actually having something to do with the class, I need to take better notes. I am having a bit of trouble with 7-1 through, well, everything but 7-4.
I get most of the sine and cosine functions. I understand the unit circle and trig chart, for the most part, but I don't have the formulas down for the word problems and whatnot that was shown to us in class.
Getting back on track, lets talk about something simple, like converting degrees to radians and vice versa!
This is clearly one of the more simple, if not the simplest, thing we will be doing in this class all year. :'(
No one else should have trouble with this if I can understand it.
All you need to know are two easy little formulas. I'll start with Degrees to Radians, give an example and then do the same with Radians to Degrees.
Degrees to Radians = pi/180
Ex. Convert 72 degrees to radians.
Instead of plugging it into your calculator just put the function down on paper and solve by treating pi as a variable.
So you would proceed with 72/1 x pi/180
Multiply straight across 72 x pi
1 x 180
And your product comes out to be 72 pi/180
That reduces to 2 pi/5
And that is your answer, if you can't reduce in your head just plug the fraction into the calculator without the pi in the function.
On the Flip side!
If you need to convert Radians to Degrees you just use the reciprocal of the first formula.
Radians to Degrees = 180/ pi
Ex. Convert 4 pi/ 2 to degrees
You set up the equation 4 pi x 180
2 x pi
Now you can clearly see that you can cancel out the bottom half of the equation.
The pi's cancel out and since 180 is divisible by 2 it can be reduced.
You are then left with 4 x 90
Your answer is then 360 Degrees
That is it for tonight's lesson, I sincerely hope that it helped someone in some way.
Malorie
This week, we learned about reference angles in 7-4. I found this was the simplest concept to understand. Most of the stuff we learned this week was pretty easy and i found it was alot easier after i did the homework.
To find a reference angle for sin and cos you must follows these steps:
step 1: figure out the which quadrant the angle is in. (if given a fraction with pi, use pi as 180 and find your angle)
step 2: determine whether the function is positive or negative.
step 3: Subtract 180 from the angle measure until it is between 0 and 90 degrees.
step 4: If the angle comes out to be 0, 30, 45, 60, or 90 degrees, plug it into the trig chart. If it is not on the trig chart then leave it as is or plug it into your calculator.
(*HINT: 0 degrees= 0; 30 degrees- pi/6; 45 degrees- pi/4; 60 degrees- pi/3; 90 degrees- pi/2*)
Ex:
Find the reference angle for sin 6pi/5
sin (6 x 180)/5 = 1080/5 = 216
(treat pi as 180)
-sin 216
this angle would be in the III quadrant because it is between 180 and 270
it would also be negative because y is negative in this quadrant.
216-180= 36
(subtract from 180)
So, your final answer would be -sin36 degrees.
This week, we also learned the trig chart. The trig chart is very helpful.
To find a reference angle for sin and cos you must follows these steps:
step 1: figure out the which quadrant the angle is in. (if given a fraction with pi, use pi as 180 and find your angle)
step 2: determine whether the function is positive or negative.
step 3: Subtract 180 from the angle measure until it is between 0 and 90 degrees.
step 4: If the angle comes out to be 0, 30, 45, 60, or 90 degrees, plug it into the trig chart. If it is not on the trig chart then leave it as is or plug it into your calculator.
(*HINT: 0 degrees= 0; 30 degrees- pi/6; 45 degrees- pi/4; 60 degrees- pi/3; 90 degrees- pi/2*)
Ex:
Find the reference angle for sin 6pi/5
sin (6 x 180)/5 = 1080/5 = 216
(treat pi as 180)
-sin 216
this angle would be in the III quadrant because it is between 180 and 270
it would also be negative because y is negative in this quadrant.
216-180= 36
(subtract from 180)
So, your final answer would be -sin36 degrees.
This week, we also learned the trig chart. The trig chart is very helpful.
Dylan's Blog...
In Advanced Math we started off with basic trig.
I understood most of it. (I'm more logic based than math.)
We learned to convert degrees to radians and vice versa.
I understood this the best...because it was the first concept.
degrees * pi/180
radians * 180/pi
Ex.
75 degrees * pi/180
(using pi as a variable of sorts)
75/180 = 5(pi)/12
(remember to treat pi like a variable)
6(pi) * 180/pi
1080 degrees (pi cancels)
Some of the other things we've learned are in the form of word problems and formula based.
I understood most of it. (I'm more logic based than math.)
We learned to convert degrees to radians and vice versa.
I understood this the best...because it was the first concept.
degrees * pi/180
radians * 180/pi
Ex.
75 degrees * pi/180
(using pi as a variable of sorts)
75/180 = 5(pi)/12
(remember to treat pi like a variable)
6(pi) * 180/pi
1080 degrees (pi cancels)
Some of the other things we've learned are in the form of word problems and formula based.
s = arc length
r = radius
Θ = radians or degrees
k = area
r = radius
Θ = radians or degrees
k = area
Formulas:
s=rΘ
k=1/2rs
k=1/2r^2Θ
Ex.
Given:
r=36cm
Θ=3(pi)/2
s=?=
s=36(Θ)
s=108/2
s=54cm
By working a problem like this, you can learn to work out word problems with apparent size(Θ).
We also learned about the trig. chart. The trig. chart allows you to convert certain degrees to radians to a number.
I also understood this the best.
Ex.
sin 60 degrees = pi/3 = √3/2
(a problem like this needs a number, not a degree, except when stated)
Overall I learned more about trig. than I already knew. I grasped the concept of the unit circle, the trig. chart, and the entire chapter. Let's hope this test will be easy.
s=rΘ
k=1/2rs
k=1/2r^2Θ
Ex.
Given:
r=36cm
Θ=3(pi)/2
s=?=
s=36(Θ)
s=108/2
s=54cm
By working a problem like this, you can learn to work out word problems with apparent size(Θ).
We also learned about the trig. chart. The trig. chart allows you to convert certain degrees to radians to a number.
I also understood this the best.
Ex.
sin 60 degrees = pi/3 = √3/2
(a problem like this needs a number, not a degree, except when stated)
Overall I learned more about trig. than I already knew. I grasped the concept of the unit circle, the trig. chart, and the entire chapter. Let's hope this test will be easy.
Charlie's Blog
In sections 7-1 through 7-6 that we have learned so far, I understood most of them. Some things took a little further explanation though. One thing I understood very well was the Sector's of a Circle (I just wasn't very good at remembering the formulas). The concept of this is to use one or more of the three formulas that we are given: k=1/2r^2theta, s=rtheta, and k=1/2rs. The problem tells you whether you have to solve to find the arc length (s), the radius (r), the central angle (theta), or the area (k). Some problems have you solve for one of these by giving you the other measurements, other problems have you solve for two.
**Some helpful hints for word problems: - diameter is another word for the arc length (s) - the apparent size is another name for the central angle (theta) - distance between the two objects is another name for the radius (r)
For Example:
The orange's apparent size is 3/7 rads. with an diameter of 7 cm. Find the distance between the orange and the bowl of fruit that's on the table.
theta= 3/7 rads.
s= 7 cm
r= ?
7 = r(3/7)
Divide 7 by 3/7 and you get that r = 16.3333
So the distance between the bowl and the orange is 16.3333 cm
In this problem it gave you the diameter (arc length), the apparent size (central angle), and asked you to solve for the distance (radius). By dividing the apparent size by the diameter, you got the distance between the two objects.
Mary's Blog
I understood most of everything B-Rob taught this week. The first thing we learned in 7-1 was how to convert degrees to radians and radians to degrees.
Example: 45 degrees
45 X pi/180 = 45pi/180 = pi/4
Example: pi/6
pi/6= 180/pi= 30 degrees
Okay, I'm gonna try and recap the harder stuff because I'm thinking it will only help me remeber how to do it.
We have to know these triples:
(3,4,5)
(5,12,13)
(7,24,25)
(8,15,17)
(9,40,41)
(11,60,61)
(12,35, 37)
(13, 84, 85)
7-4
This section was dealing with reference angles
In 7-4, the work wasn't hard, it just took alot of thinking because I was trying to remember that:
0 degrees=0
30 degrees =pi/6
45 degrees = pi/4
60 degrees= pi/3
90 degrees = pi/2
It really would help too if you know the entire trig chart.
The steps:
1. Find the Quadrant
2. Determine + or -
3. Subtract 180 until b/w 0 & 90
4. Use trig chart or leave it.
1. Quadrant III
2. Negative
3. 60 degrees
4. 60 = pi/3 on trig chart
Final answer: Sin600=-sin60=square root of 3/2
I get this stuff, if anyone needs help, take note that I'm always on facebook, just ask me on there or here. I'll be happy to help.
Example: 45 degrees
45 X pi/180 = 45pi/180 = pi/4
Example: pi/6
pi/6= 180/pi= 30 degrees
Okay, I'm gonna try and recap the harder stuff because I'm thinking it will only help me remeber how to do it.
We have to know these triples:
(3,4,5)
(5,12,13)
(7,24,25)
(8,15,17)
(9,40,41)
(11,60,61)
(12,35, 37)
(13, 84, 85)
7-4
This section was dealing with reference angles
In 7-4, the work wasn't hard, it just took alot of thinking because I was trying to remember that:
0 degrees=0
30 degrees =pi/6
45 degrees = pi/4
60 degrees= pi/3
90 degrees = pi/2
It really would help too if you know the entire trig chart.
The steps:
1. Find the Quadrant
2. Determine + or -
3. Subtract 180 until b/w 0 & 90
4. Use trig chart or leave it.
Detailed Example: Find a reference anlge of cos225 degrees
First, find the quadrant, 225 is between 180 and 270 so its in the third quadrant.
Cos is related to the x axis, x is negative in the third quadrant, which makes it negative
Next, start subtracting 180 until you reach an angle between 0 and 90
225-180=45 degrees, which is on the trig chart! joy!
45 degrees is pi/4, take out your trig chart and look at the cos part, look for pi/4
there's your answer! -squareroot of 2/2! (the negative is from the second step,it is carried)
Short Example: Find a reference angle of sin 600 degrees
1. Quadrant III
2. Negative
3. 60 degrees
4. 60 = pi/3 on trig chart
Final answer: Sin600=-sin60=square root of 3/2
I get this stuff, if anyone needs help, take note that I'm always on facebook, just ask me on there or here. I'll be happy to help.
Saturday, August 28, 2010
Tori's Blog
This week, we covered 7-1 through 7-4. I understood most of this weeks lessons, however, 7-3 bothered me the most. It wasn't that it was hard or anything, it was that it was a lot of work. Although, I learned that if we memorized the Pythagorean Triples, it would be easy to solve them.
-- some examples of Pythagorean Triples:
-3,4,5
-5,12,15
-7,24,25
-8,15,17
-9,40,41
-11,60,61
-12,35,37
-13,84,85
I also had trouble with parts of 7-1. Converting from radians to degrees and degrees to radians was very easy.
rads-degrees: x 180/pi degrees-rads: x pi/180
The main thing in 7-1 I did not understand was finding coterminal angles when it was in radians, but now I know that all you do is subtract 2pi from the rad and you get your answer.
7-2 was my favorite lesson because it was the lesson I understood the most. the formulas...:
-s=r0 -k=1/2 r^2 0 -k=1/2 rs
--s=arc length r=radius 0=central angle k=area of a sector
...were very easy to understand and plugging in the numbers were quick and easy.
7-4 was easy to understand but having to look into the trig chart to check for the angle got old very quick.
-- some examples of Pythagorean Triples:
-3,4,5
-5,12,15
-7,24,25
-8,15,17
-9,40,41
-11,60,61
-12,35,37
-13,84,85
I also had trouble with parts of 7-1. Converting from radians to degrees and degrees to radians was very easy.
rads-degrees: x 180/pi degrees-rads: x pi/180
The main thing in 7-1 I did not understand was finding coterminal angles when it was in radians, but now I know that all you do is subtract 2pi from the rad and you get your answer.
7-2 was my favorite lesson because it was the lesson I understood the most. the formulas...:
-s=r0 -k=1/2 r^2 0 -k=1/2 rs
--s=arc length r=radius 0=central angle k=area of a sector
...were very easy to understand and plugging in the numbers were quick and easy.
7-4 was easy to understand but having to look into the trig chart to check for the angle got old very quick.
Lawrence's Blog
The concept i understood best from this week was from 7-1.
Converting degrees to rads and rads to degrees.
To convert degrees to rads, its degree X pi/180
Ex. 45 degrees
45 X pi/180= 180/6= pi/4.
Ex. for converting rads to degrees.
pi/6
pi/6 X 180/pi= 180/6 = 30 degrees
remember that the pi's cancel out!!!
I also understood how to get positive and negative coterminal angles.
if its in degrees you substract or add from 360. if its a large negative number and you need to get a positive coterminal angle keep adding it from 360 till you get a positive number.
ex. -1030 degrees
-1030+360+360+360= 50 degrees. (positive)
-1030-360= -1390 degrees. (negative)
if its in radians you just add 2pi or subtract 2pi
Ex. pi/6
pi/6+ 2pi= 13pi/6 (positive)
pi/6- 2pi= -11pi/6 (negative)
The other concept i understood was from 7-4
How to find a reference angle.
1. find the quadrant the angle is in
2. determine if the trig function is positive or negative
3. subtract 180 degrees from the angle until theta is b/w 0 and 90 degrees
4. if it is a trig chart angle plug in. if not leave it or plug in calculator.
ex. cos 225 degrees
its in quad 3
its negative (-cos)
225-180= 45 -cos 45 degrees. (trig chart function)
= -square root 2/2
that is all the concepts that i know the best the others i am decent with so if you need help ask me in class and i will be glad to help.
Converting degrees to rads and rads to degrees.
To convert degrees to rads, its degree X pi/180
Ex. 45 degrees
45 X pi/180= 180/6= pi/4.
Ex. for converting rads to degrees.
pi/6
pi/6 X 180/pi= 180/6 = 30 degrees
remember that the pi's cancel out!!!
I also understood how to get positive and negative coterminal angles.
if its in degrees you substract or add from 360. if its a large negative number and you need to get a positive coterminal angle keep adding it from 360 till you get a positive number.
ex. -1030 degrees
-1030+360+360+360= 50 degrees. (positive)
-1030-360= -1390 degrees. (negative)
if its in radians you just add 2pi or subtract 2pi
Ex. pi/6
pi/6+ 2pi= 13pi/6 (positive)
pi/6- 2pi= -11pi/6 (negative)
The other concept i understood was from 7-4
How to find a reference angle.
1. find the quadrant the angle is in
2. determine if the trig function is positive or negative
3. subtract 180 degrees from the angle until theta is b/w 0 and 90 degrees
4. if it is a trig chart angle plug in. if not leave it or plug in calculator.
ex. cos 225 degrees
its in quad 3
its negative (-cos)
225-180= 45 -cos 45 degrees. (trig chart function)
= -square root 2/2
that is all the concepts that i know the best the others i am decent with so if you need help ask me in class and i will be glad to help.
Friday, August 27, 2010
Nathan's Blog
The concept that I understood the best this week was from 7-1.
Converting degrees to radians: degrees X PI/180
Ex. 120 degrees to radians= 120 X PI/180= 120PI/180= 2PI/3
Converting radians to degrees: radians X 180/PI
Ex. PI/15 to degrees= PI/15 X 180/PI= 180/15= 12 degrees
The other concept that I really understood was reference angles.
1.) Determine which quadrant that angle is in.
2.) Figure out if the function is positive or negative.
3.) Subtract 180 degrees from that certain angle until its absolute value is between 0 & 90
4.) If you end up with a trig chart angle, plug it in. If you didn't, then leave it alone or plug it into your calculator.
Here are the trig chart angles: 0 degrees= 0, 30 degrees= PI/6, 45 degrees= PI/4,
60 degrees= PI/3, and 90 degrees= PI/2.
Ex. sin 520 (HINT: You can subtract 360 from the given angle to get it between 0 & 360 so that it is easier to determine which quadrant the angle is in.)
520-360=160, so it is in Quadrant II.
sin is related to the Y axis, so in Q II, sin is positive.
Now you must subtract 180 from 520 until the absolute value is b/w 0 & 90.
520-180-180-180= -20
Since you want absolute value, you ignore the negative.
So your final answer is: sin20 degrees.
Those were the best two concepts that I understood, but the word problems were pretty easy too. The five trig functions were understandable, but it took me a while to get used to working them. So, for the most part, advanced math is somewhat easy, but then it isn't. This blog was also easier than I thought. This will probably be the easiest week of math this year, because it is bound to get tougher. BYE
Converting degrees to radians: degrees X PI/180
Ex. 120 degrees to radians= 120 X PI/180= 120PI/180= 2PI/3
Converting radians to degrees: radians X 180/PI
Ex. PI/15 to degrees= PI/15 X 180/PI= 180/15= 12 degrees
The other concept that I really understood was reference angles.
1.) Determine which quadrant that angle is in.
2.) Figure out if the function is positive or negative.
3.) Subtract 180 degrees from that certain angle until its absolute value is between 0 & 90
4.) If you end up with a trig chart angle, plug it in. If you didn't, then leave it alone or plug it into your calculator.
Here are the trig chart angles: 0 degrees= 0, 30 degrees= PI/6, 45 degrees= PI/4,
60 degrees= PI/3, and 90 degrees= PI/2.
Ex. sin 520 (HINT: You can subtract 360 from the given angle to get it between 0 & 360 so that it is easier to determine which quadrant the angle is in.)
520-360=160, so it is in Quadrant II.
sin is related to the Y axis, so in Q II, sin is positive.
Now you must subtract 180 from 520 until the absolute value is b/w 0 & 90.
520-180-180-180= -20
Since you want absolute value, you ignore the negative.
So your final answer is: sin20 degrees.
Those were the best two concepts that I understood, but the word problems were pretty easy too. The five trig functions were understandable, but it took me a while to get used to working them. So, for the most part, advanced math is somewhat easy, but then it isn't. This blog was also easier than I thought. This will probably be the easiest week of math this year, because it is bound to get tougher. BYE
Kaitlyn's Blog
This week we learned about the Trigonometric Functions, which are:
1. Sine
2. Cosine
3. Tangent
4. Cotangent
5. Secant
6. Cosecant
Sine, in the coordinate plane, is defined as y/r. It is positive in the quadrants where y is the positive. Cosine is defined as x/r. It is positive in the quadrants where x is the positive. To find Sin (theta) and Cos (theta) of a circle in the unit circle, you would take the point given and mark it on the coordinate plane. Then you would connect the points to form a triangle.
Ex: If the terminal ray of an angle in standard position passes through ( -4, 6), find sin(theta) and cos(theta).
-To solve this you must mark the point on the coordinate plane, draw a line from the point going down and another line from the point to the origin. Then find the hypoteneuse (r).
- sin(theta)=y/r cos(theta)=x/r
Tangent: tan(theta)=y/x
Cotangent: cot(theta)=x/y
Secant: sec(theta)=r/x
Cosecant: csc(theta)=r/y
To solve for these equations, you would do the same thing that you did to solve for Sine and Cosine.
I understood this week's lessons pretty well, but sometimes it could get confusing. We also learned about solving reference angles. This will only be complicated for me because of the trig chart. It's a lot to remember. I got confused on how to solve the inverses of the functions without a calculator. I'm still not quite sure how to do it, but i think that i will be able to get the hang of it.
1. Sine
2. Cosine
3. Tangent
4. Cotangent
5. Secant
6. Cosecant
Sine, in the coordinate plane, is defined as y/r. It is positive in the quadrants where y is the positive. Cosine is defined as x/r. It is positive in the quadrants where x is the positive. To find Sin (theta) and Cos (theta) of a circle in the unit circle, you would take the point given and mark it on the coordinate plane. Then you would connect the points to form a triangle.
Ex: If the terminal ray of an angle in standard position passes through ( -4, 6), find sin(theta) and cos(theta).
-To solve this you must mark the point on the coordinate plane, draw a line from the point going down and another line from the point to the origin. Then find the hypoteneuse (r).
- sin(theta)=y/r cos(theta)=x/r
Tangent: tan(theta)=y/x
Cotangent: cot(theta)=x/y
Secant: sec(theta)=r/x
Cosecant: csc(theta)=r/y
To solve for these equations, you would do the same thing that you did to solve for Sine and Cosine.
I understood this week's lessons pretty well, but sometimes it could get confusing. We also learned about solving reference angles. This will only be complicated for me because of the trig chart. It's a lot to remember. I got confused on how to solve the inverses of the functions without a calculator. I'm still not quite sure how to do it, but i think that i will be able to get the hang of it.
Taylor's Blog
In the following lesson I shall be informing you on lesson 7.1 of my Advanced Math class.
There are two units of measure for angles. They are degrees and radians.
If you want to convert degrees to radians you must use the following formula.
degrees x PI/180
An example of this would be the following problem.
Convert 15 degrees to radians.
15 degrees x PI/180 1st. Place the degrees ,15, into its proper location in the equation.
15PI/180 You will then multiply 15 degrees and PI/180, or 15 X PI to get this answer.
PI/12 You will then divide 180 into 15,or reduce, and put it in a fraction form. This is your answer.
IMPORTANT NOTE: PI IS LIKE X A VARIABLE SO DON'T PUT IT INTO YOUR CALCULATOR OR YOU WILL GET THE WRONG ANSWER!
If you want to convert radians to degrees you would use this formula.
rads x 180/PI
An example of this equation has been given.
Convert 12PI to degrees.
12PI X 180/PI First place your radians into its proper place in the equation.
12P/I X `180/P/I Since 12PI and 180/PI both have PI and one is in the lower part of the fraction and the other is in the upper part of the fraction. We can cancel out PI in the equation.
12 X 180 You will then multiply 12 times 180.
2160 degrees This is what your answer should be.
IMPORTANT NOTE: MAKE SURE TO PUT THE DEGREES MARK OR IT WILL BE WRONG!!
There are some angles that are called coterminal angles. Coterminal means that the degrees 'spins' around the angle in this usage.
If you are asked to find the positive or/and negative form of a coterminal angle for degrees you will use this formula.
degrees +/- 360 degrees
Yet, if you are asked to find the positive or/and negative form of a coterminal angle for radians you would use this formula.
rads +/- 2 PI
An example of these situations are as followed.
Find a positive coterminal angle of 12 degrees.
12 degrees = 360 degrees First, you will place 12 degrees into the equation and since we need it to be positive we will add 360 to it.
372 degrees This is the correct answer that you should have gotten after adding 360 to 12.
Find the negative coterninal angle of 18 degrees.
18 degrees - 360 degrees First place 18 into its correct place in the equation and since we need to find the negative coterminal angle we will subtract 360 from 18 instead of adding it.
-342 degrees This is the negative answer that you should have gotten.
Now lets find the positive coterminal angle for 4PI/6.
4PI/6 + 2PI First place the 4PI/6 in its correct location in the equation and since we need to find the positive coterminal angle you will add 2Pi.
4pi/6 + 12pi/6 Now convert the 2PI so that you can add correctly. Do this by multiplying 2 by 6 and placing a six in the denominator.
16PI/6 After you do that you add the fractions together.
8PI/3 Then you reduce the fraction by dividing it by two.
Now lets find the negative coterminal angle of 8PI/6.
8PI/6 - 2PI First place 8Pi/6 in its proper location in the equation and since we are finding the negative coterminal angle you will subtract 2PI instead of adding it.
8PI/6 - 12PI/6 Than you will make 2PI into a fraction by placing it over 6 and multiplying it by 6.
-4PI/6 You will then subtract and get this answer.
-2PI/3 Then you will reduce the fraction by diving it by two and this is your answer.
Now lets learn how to get minutes and seconds from our equations.
ab.cd
.cd x 60 = y minutes y'
y x60 = z seconds z''
This is the formula that we will be using in the following problem.
Covert 15.699 degrees to minutes and seconds.
.699 x 60= 41.94 First we will put .699 into the equation because it is after the decimal point. We will then multiply it by 60 so that we can get 41.94. Since we have another decimal point we must continue the equation.
.94 x 60= 56.4 We will take the numbers after the decimal and multiply them by 60. Since we don't need the .4 you will simply drop it, since we just need seconds.
15 degrees 41' 56'' This is what your answer should be and what it should look like.
The final thing that you must know is how to convert the minutes and seconds back into degrees. You will use the following formula in order to do this.
x'=minutes y''= seconds
x/60 = y/3600= degrees
Now use it to solve the following problem.
Convert 10 degrees 56' 78'' in to degrees.
56/60 = 78/3600 First place the seconds and minutes in their appropriate location in the formula.
.955 Then you add the fractions together and convert the fraction that you may have gotten if you are doing this by hand into a decimal.
10.955 degrees Than you simply place the decimal behind the number and that is how you get your answer.
That is all the equations, formula and information that we learned in section 7.1. Until my next blog BYE.
There are two units of measure for angles. They are degrees and radians.
If you want to convert degrees to radians you must use the following formula.
degrees x PI/180
An example of this would be the following problem.
Convert 15 degrees to radians.
15 degrees x PI/180 1st. Place the degrees ,15, into its proper location in the equation.
15PI/180 You will then multiply 15 degrees and PI/180, or 15 X PI to get this answer.
PI/12 You will then divide 180 into 15,or reduce, and put it in a fraction form. This is your answer.
IMPORTANT NOTE: PI IS LIKE X A VARIABLE SO DON'T PUT IT INTO YOUR CALCULATOR OR YOU WILL GET THE WRONG ANSWER!
If you want to convert radians to degrees you would use this formula.
rads x 180/PI
An example of this equation has been given.
Convert 12PI to degrees.
12PI X 180/PI First place your radians into its proper place in the equation.
12P/I X `180/P/I Since 12PI and 180/PI both have PI and one is in the lower part of the fraction and the other is in the upper part of the fraction. We can cancel out PI in the equation.
12 X 180 You will then multiply 12 times 180.
2160 degrees This is what your answer should be.
IMPORTANT NOTE: MAKE SURE TO PUT THE DEGREES MARK OR IT WILL BE WRONG!!
There are some angles that are called coterminal angles. Coterminal means that the degrees 'spins' around the angle in this usage.
If you are asked to find the positive or/and negative form of a coterminal angle for degrees you will use this formula.
degrees +/- 360 degrees
Yet, if you are asked to find the positive or/and negative form of a coterminal angle for radians you would use this formula.
rads +/- 2 PI
An example of these situations are as followed.
Find a positive coterminal angle of 12 degrees.
12 degrees = 360 degrees First, you will place 12 degrees into the equation and since we need it to be positive we will add 360 to it.
372 degrees This is the correct answer that you should have gotten after adding 360 to 12.
Find the negative coterninal angle of 18 degrees.
18 degrees - 360 degrees First place 18 into its correct place in the equation and since we need to find the negative coterminal angle we will subtract 360 from 18 instead of adding it.
-342 degrees This is the negative answer that you should have gotten.
Now lets find the positive coterminal angle for 4PI/6.
4PI/6 + 2PI First place the 4PI/6 in its correct location in the equation and since we need to find the positive coterminal angle you will add 2Pi.
4pi/6 + 12pi/6 Now convert the 2PI so that you can add correctly. Do this by multiplying 2 by 6 and placing a six in the denominator.
16PI/6 After you do that you add the fractions together.
8PI/3 Then you reduce the fraction by dividing it by two.
Now lets find the negative coterminal angle of 8PI/6.
8PI/6 - 2PI First place 8Pi/6 in its proper location in the equation and since we are finding the negative coterminal angle you will subtract 2PI instead of adding it.
8PI/6 - 12PI/6 Than you will make 2PI into a fraction by placing it over 6 and multiplying it by 6.
-4PI/6 You will then subtract and get this answer.
-2PI/3 Then you will reduce the fraction by diving it by two and this is your answer.
Now lets learn how to get minutes and seconds from our equations.
ab.cd
.cd x 60 = y minutes y'
y x60 = z seconds z''
This is the formula that we will be using in the following problem.
Covert 15.699 degrees to minutes and seconds.
.699 x 60= 41.94 First we will put .699 into the equation because it is after the decimal point. We will then multiply it by 60 so that we can get 41.94. Since we have another decimal point we must continue the equation.
.94 x 60= 56.4 We will take the numbers after the decimal and multiply them by 60. Since we don't need the .4 you will simply drop it, since we just need seconds.
15 degrees 41' 56'' This is what your answer should be and what it should look like.
The final thing that you must know is how to convert the minutes and seconds back into degrees. You will use the following formula in order to do this.
x'=minutes y''= seconds
x/60 = y/3600= degrees
Now use it to solve the following problem.
Convert 10 degrees 56' 78'' in to degrees.
56/60 = 78/3600 First place the seconds and minutes in their appropriate location in the formula.
.955 Then you add the fractions together and convert the fraction that you may have gotten if you are doing this by hand into a decimal.
10.955 degrees Than you simply place the decimal behind the number and that is how you get your answer.
That is all the equations, formula and information that we learned in section 7.1. Until my next blog BYE.
This week i didnt really undersand alot of the material we learned. The sine and cosine functions were some of the things that i struggled with. The formulas was one of the other things i struggled with to remember for the quizes we took. Some of the other things were the tangent, cotangent, secant, and cosecant. I struggled with trying to figure out what quadrant they belonged in, so i couldnt solve the problem. Some of the things that i understood was the measurement of angles. That was probaly the easiest thing i learned this year. Trying to find the inverse of the trig functions was pretty difficult to do. Most of the stuff i learned this week i didnt really understand.
Nicala's 1st Blog
On this blog I am going to explain the relationships between the following functions:
Sine and Cosine
Tangent and Cotangent
Sine and Cosecant,
Cosine and Secant
Sine, Cosine & Tangent.
In a right triangle, there are three sides and two non-right angles. All triangle have a adjacent side, opposite side, and hypotenuse.
Opposite is opposite to the angle θ.
Adjacent is adjacent (next to) to the angle θ.
Hypotenuse is the long one.
Sine is the length of the side opposite to an angle and the length of the triangle's hypotenuse and cine is the length of the side adjacent to an angle and the length of the triangle's hypotenuse.
Tangent and cotangent are both the length of the side opposite to an angle and the side adjacent to the angle.
Cosine and secant are the length adjacent side to an angle and the length of the triangle's hypotenuse.
Sine, cosine,and tangent are related because when put them together they form a right triangle.
Sine and Cosine
Tangent and Cotangent
Sine and Cosecant,
Cosine and Secant
Sine, Cosine & Tangent.
In a right triangle, there are three sides and two non-right angles. All triangle have a adjacent side, opposite side, and hypotenuse.
Opposite is opposite to the angle θ.
Adjacent is adjacent (next to) to the angle θ.
Hypotenuse is the long one.
Sine is the length of the side opposite to an angle and the length of the triangle's hypotenuse and cine is the length of the side adjacent to an angle and the length of the triangle's hypotenuse.
Tangent and cotangent are both the length of the side opposite to an angle and the side adjacent to the angle.
Cosine and secant are the length adjacent side to an angle and the length of the triangle's hypotenuse.
Sine, cosine,and tangent are related because when put them together they form a right triangle.
Thursday, August 26, 2010
Wednesday, August 25, 2010
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