Reflection

Well this week ill post some stuff about logs because its something simple, so i can not be so stressed about trig. I'll just give some examples of what i thought was the easiest thing we learned this year: logs.

EXAMPLES:
Condensing.

1.)logm + log7 + 4logn
= log7mn^4

2.)5loga + logd + log6
= log6da^5

3.)logn - 3logh -logy
= n/yh^3

4.)4logt - logc
= t^4/c

Expanding.

1.)log5gh^2
= log5 + 2logh +logg

2.)m^3b^7/f
= 3logm + 7logb - logf

Reflection #37

Again, I'm going to review some trig that's going to be on next weeks Trig Exam:

First, all of the formulas we learned:
** a = alpha B = beta

cos (a+B) = cos a cos B - sin a sin B
cos (a-B) = cos a cos B + sin a sin B
sin (a+B) = sin a cos B + cos a sin B
sin (a-B) = sin a cos B - cos a sin B
sin x + sin y = 2sin (x+y/2) cos (x-y/2)
sin x - sin y = 2cos (x+y/2) sin (x-y/2)
cos x + cos y = 2cos(x+y/2) cos (x-y/2)
cos x - cos y = -2sin (x+y/2) sin(x-y/2)
tan (a+B) = tan a + tan B/1-tan a tan B
tan (a-B) = tan a - tan B/1+tan a tan B
sin 2a = 2sin a cos a
cos 2a = cos^2 a - sin^2 a
cos 2a = 1-2sin^2 a
cos 2a = 2cos^2 a -1
tan 2a = 2tan a/1-tan^2 a
sin a/2 = +/- square root of 1-cos a/2
cos a/2 = +/- square root of 1+cos a/2
tan a/2 = +/- square root of 1-cos a/1+cos a
tan a/2 = sin a/1+cos a
tan a/2 = 1-cos a/sin a

Ex: Find the exact value of sin 15 degrees
a=45 degrees B=30 degrees
sin (a-B) = sin a cos B - cos a sin B
sin (45-30) = sin 45 cos 30 - cos 45 sin 30
sin 15 = (square root of 2 over 2)(square root of 3 over 2) - (square root of 2 over 2)(1/2)
sin 15 degrees = (square root of 6 over 4) - (square root of 2 over 4)
= square root of 6 - square root of 2 all over 4

If tan a = 2 and tan B = -1/3 then find tan (a+B)
tan (a+B) = tan a + tan B/1-tan a tan B
tan (a+B) = 2 + -1/3 / 1-(2 x -1/3)
tan (a+B) = 5/3 / 5/3
= 1

2cos^2 10 -1
Use...cos 2a = 2cos^2 a -1
cos 2(10)
=cos 20 degrees

Now, conversions:

To convert degrees to radians: multiply by pi/180
ex. 120 degrees
120 times pi/180
(keep answer in pi form)
120 divided by 180
=2/3
your answer =2/3 pi or 2 pi/3

To convert radians to degrees: multiply by 180/pi
ex. 3 pi/4 or 3/4 pi
3 pi/4 times 180/pi
pi cancels out
3/4 times 180
3 times 180
=540
divided by 4
your answer =135 degrees

To convert degrees to radians, it has to be in decimal form:
119.2
37.92
and so forth.
119.2/180 (plug in cal.) 149/225
then multiply pi =149 pi/225

To convert degrees decimals to minutes and seconds:
take the decimal of the degrees and multiply by 60 for minutes
68.33 .33 times 60 =19.8
take the decimal of the minutes and multiply by 60 for seconds
19.8 .8 times 60 =48
your answer =68 degrees 19 minutes 48 seconds

*if seconds have decimal, round to the 3rd decimal place
*if minutes don't have decimals, leave in degrees minutes

To convert degrees minutes seconds to decimals:
68 degrees 19 minutes 48 seconds
divide seconds (48) by 3600 =.013
divide minutes (19) by 60 =.317
add the two decimals =.33
your answer is 68.33

heres another late blog. its a review from chapter 13. Find the sum of the multiples of 3 between 1 and 1000*The first thing you want to do is find the smallest and largest multiple of three between those numbers.*The smallest multiple is 3 and the largest is 999*You can use the arithmetic formula for this problem like this:999=3+(n-1)(3)999=3nn=333So now that you have 'n' you can plug all your numbers into the sum formula to get the answer*So you get Sn=333(3+999)/2And that gives you 166833

late blog

this is one of my make up blogs. so heres somthing that im going to review from chapter 10. Simplify tan75-tan30/(1+tan75tan30)*When you first look at this problem you should notice that it is a formula*It's a formula for tan(a-b) **a=alpha;b=beta*So all you do is plug the two numbers you're given into tan(a-b) where the bigger number is alpha and the other number is beta (So a=75 and b=30)*So you get tan(75-30)= tan 45And they're asking you to simplify, so that means your answer should be a number*tan 45 is 1, so 1 is your answer

Topics

Write a blog on one of the following

1. Explain in detail the steps to graphing a trig function

2. Explain how to solve a non-right triangle with Law of Sines and Law of Cosines

3. Explain how to know what type of polar graph you have given an equation

Reflection #36

I'm going to review the trig we've been working on in class:

First, the trig chart:
0 = 0 degrees
pi/6 = 30 degrees
pi/4 = 45 degrees
pi/3 = 60 degrees
pi/2 = 90 degrees
sin 0 = 0
sin 30 = 1/2
sin 45 = square root of 2 over 2
sin 60 = square root of 3 over 2
sin 90 = 1
cos 0 = 1
cos 30 = square root of 3 over 2
cos 45 = square root of 2 over 2
cos 60 = 1/2
cos 90 = 0
tan 0 = 0
tan 30 = square root of 3 over 3
tan 45 = 1
tan 60 = square root of 3
tan 90 = undefined
cot 0 = undefined
cot 30 = square root of 3
cot 45 = 1
cot 60 = square root of 3 over 3
cot 90 = 0
sec 0 = 1
sec 30 = 2 square root of 3 over 3
sec 45 = square root of 2
sec 60 = 2
sec 90 = undefined
csc 0 = undefined
csc 30 = 2
csc 45 = square root of 2
csc 60 = 2 square root of 3 over 3
csc 90 = 1

Examples:
**a=alpha
4 sin pi/6 cos pi/6
=2 sin 2a
=2 sin 2(pi/6)
=2 sin(pi/3)
=2(square root of 3 over 3)
=square root of 3

cos^2 x/2 - sin^2 x/2
=cos 2a
=cos 2(x/2)
=cos x

Now, identities:
csc x = 1/sin x
sec x = 1/cos x
cot x = 1/tan x
sin (-x) = -sin x
cos (-x) = cos x
csc (-x) = -csc x
sec (-x) = sec x
tan (-x) = -tan x
cot (-x) = -cot x
sin^2 x + cos^2 x = 1
1 + tan^2 x = sec^2 x
1 + cot^2 x = csc^2 x
sin x = cos (90 degrees - x)
tan x = cot (90 degrees - x)
sec x = csc (90 degrees - x)
cos x = sin (90 degrees - x)
cot x = tan (90 degrees - x)
csc x = sec (90 degrees - x)
tan x = sin x/cos x
cot x = cos x/sin x

Examples:
csc x - sin x
1/sin x - sin x/1
multiply sin x/1 by sin x/sin x
1/sin x - sin^2 x/sin x
1 - sin^2 x/sin x
cos^2 x/sin x
= cot x cos x

cot x + tan x
cos x/sin x + sin x/cos x
multiply cos x/sin x by cos x/cos x
multiply sin x/cos x by sin x/sin x
cos^2 x/cos x sin x + sin^2 x/cos x sin x
cos^2 x + sin^2 x/cos x sin x
1/cos x sin x
1/cos x times 1/sin x
= csc x sec x

sin x cot x
sin x/1 times cos x/sin x
sin x cancels out
= cos x

reflection

Cos(α +/- β)=cos α cos β -/+ sin α sin β
sin(α +/- β)=sin α cos β -/+ cos α sin β
sin x + sin y= 2 sin x + y/2 cos x-y/2
sin x - sin y= 2 cos x + y/2 sin x-y/2
cos x + cos y= 2 cos x + y/2 cos x-y/2
cos x - cos y= 2 sin x + y/2 sin x-y/2

tan (α + β)=tan α + tan β/1-tan α tan β
tan (α - β)=tan α - tan β/1+tan α tan β

sin2α=2sin α cos α
cos 2α=cos^2 α –sin^2 α = 1-2 sin^2 α= 2 cos^2 α -1
tan 2α = 2tan α /1-tan^2 α
sin α/2= +/- √1-cos α/2
cos α/2= +/- √1+ cos α/2
tan α/2= +/- √1-cos α or 1 + cos α
=sin α/1+cos α
=1-cos α/sin α

tan α = 2 and tan β=1
find tan (α - β)
= tan α + tan β/1-tan α tan β
=2+1/1-(2)(6)
=3/-1
=-3

Find the exact value of: tan 15+tan 30/1-tan 15 tan 30
tan α = 2 and tan β=1
find tan (α - β)
= tan (15 + 30)
=tan (45)
=1

Reflection

Solving anything bigger than a quadratic usuing quadratic form. To use quadratic form you have to have 3 terms only. The first term must equal the 2nd exponentx2, and the last term must be a constant. The first thing you do is make g=x^exponent/2 so that you would get g^2+g+#. The second thing is to do the quadratic formula, factor, or complete the square. The the last thing is to plug back in for g. (Whenever you do step three you are basically just plugging back into step one. g=x^2)

An example:
x^4-4x^2-12=0

1. g=x^4/2
g=x^2
g^2-4g-12

2. (g^2-6g)+(2g-12)
g(g-6)+2(g-6)
(g+2)(g-6)
g=-2 g=6

Triangle Formulas:

A= 1/2 (leg) (leg) Sin (angle between). This is used for non-right triangles.

For Area say you have a triangle with:

A side length of 4
A side length of 5
And an angle of 30 degrees

Then plug it all in: A=1/2 (4) (5) Sin 30 degrees

A= 10 Sin 30 degrees which aproximately = 5.


For another triangle:

A side length of 3
A side length of 8
And an angle of 60 degrees

Then plug it all in: A= 1/2 (3) (8) Sin 60 degrees

A= 12 Sin 60 degrees which aproximately = 10.392

This formula can only be done when you have two given lengths and at least one angle.

4/26

The Dot Product is something that we just learned in advanced math.
Here are the formulas and how you do it.

DOT PRODUCT:
(2,3) (1,4)
**Formula v1v2=x1(x2) + y1(y2)
Multiply:
2x1=2 + 3x4=12
2 + 12 = 14

MIDPOINT:formula m = (x1+x2/2, y1+y2/2, z1+z2/2)
Find the midpoint of (2,2,2) and (2,4,6)
2+2/2, 2+4/2, 2+6/2
=(2, 3, 4)

SPHERE:
A(8,-2,3) and B(4,0,7) are endpoints of a diameter. Find the equation of the sphere.

**MIDPOINT
C=(8+4/2, -2+0/2, 3+7/2)
C=(6,-1,5)

Reflection

For this week's blog I'll just review some things from Chapter 10. So here's a few examples:

Ex. 1.) If sinx=12/13 Find tan x/2
*Okay for this problem they're asking you to find tan x/2, and that has a formula:
sinx/(1+cos x) So you're going to use that formula to find the answer
*So first you have to draw a triangle from what you're given (sinx=12/13)
(So the height of the triangle is 12, the hypotenuse is 13, and the base is 5)
*Now you can plug into the formula:
tan x/2 = sin x/(1+cos x)
=12/13 / 1+(5/13)
= 12/13 / 18/13 then sandwich it
=156/234 = 2/3
*So tan x/2 is 2/3

Ex. 2.) Simplify tan75-tan30/(1+tan75tan30)
*When you first look at this problem you should notice that it is a formula
*It's a formula for tan(a-b) **a=alpha;b=beta
*So all you do is plug the two numbers you're given into tan(a-b) where the bigger number is alpha and the other number is beta (So a=75 and b=30)
*So you get tan(75-30)
= tan 45
And they're asking you to simplify, so that means your answer should be a number
*tan 45 is 1, so 1 is your answer

Ex. 3.) Simplify cos(60 + y) + cos(60 - y)
*For this problem you're going to have to use the cosine sum and difference formulas and you'll have to work the two parts of the problem separately
*So first you can solve cos(60 + y) Plugging that into the formula you get:
(cos60cosy - sin60siny)
^^And that simplifies to give you (1/2)cosy - sqrt of 3/2siny
*Now you can solve cos(60 - y)
So you get: (cos60cosy-sin60siny)
^And that simplifies to (1/2)cosy + sqrt of 3/2siny
Now you add everything together:
(1/2cosy - sqrt of 3/2siny + 1/2cosy + sqrt of 3/2siny
sqrt of 3/2's cancel out so you're left with 1/2cosy + 1/2cosy
And that equals 1cosy => cos y is your answer

Ex. 4.) If tan x = 1/2 and tan y = 2/3 Find tan(x+y)
*they're asking for tan(x+y) so all you have to do is plug the numbers into the tangent sum formula:
tna(a+b) = tanz+tanb/(1-tanatanb)
*so you get 1/2 + 2/3 / 1-(1/2)(2/3)
= 7/6 / 1-1/3
=7/6 / 2/3 sandwich it and you get 21/12
*So your simplified answer is 7/4

Reflection 4/25

so, i guess im awake enough to do this now XD
ill do it on the unit circle and values of the functions

Unit Circle:

90°, (0,1), π/2
360°, 0°, (1,0), 2π
180°, (-1,0), π
270°, (0,-1), 3π/2

sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y

now, knowing all of this can be verrrryyy useful

if you see something like sin(8π/5), you can use basic knowledge to kno 2 things: a) that the angle is in the 4th quadrant b) that the angle's going to have a negative y coordinate and r will be positive
now, you can use math to realize that this angle's measurement in degrees is 288°
then, you could get the reference angle, which is 72°, and then you just plug it into sin72, and then you get 0.254

well, hope this was informative, and i hope everybody had a good time at prom, i kno i did :)
o and one more thing, cant wait till tomorrow, my birthday :D
k, cya y'all

Some more review on my favorite thing in trig because its SUPER EASY.


SOHCAHTOA!!!!!!

sin=opposite leg/hypotenuse
cos=adjacent leg/hypotenuse
tan=opposite leg/adjacent leg
*You use SOCAHTOA for right triangles
Here are some examples on using SOHCAHTOA....
solve for b & c.
1) You have an ABC triangle with angle A 28 degrees, a=40, and angle C is a right triangle.(The first thing you would do is draw out this problem and label all your sides and angles if they are given. Then look at this problem and see which angles or sides go with what, sin, tan, or cos. In this problem it's tan because once you draw it out 40 is across from you angle and C is adjacent.)It's tan because
tan28=40/b
btan28=40
b=40/tan28
which equals about 75.229
sin28=40/c
c=40/sin28
which equals about 85.202

reflection april 25

This week went by extremely slow. All we did was go over our old tests and ask questions to prepare for the giant trig exam. This weekend went by pretty fast, won the second round of playoffs and going to statteee! And then rushing back home to get ready for prom which was fun. Now back to another week of school getting ready for the trig exam. Heres some stuff from chapters 7 and 8 that we did during the week.

--CHAPTER 7--

10) tan(cos-1(12/13))
for this problem you would draw a triangle and then use SOHCAHTOA
you have 13, 5, and 12 in your triangle. then you would get 5/12 as your answer because tan=opp/adj


17a) sin 626
_sin_
the first blank would be pos or neg and the second would be for the angle between 0 &90
626-360=266
266-180=86

--CHAPTER 8--

5) cosxcotx=2cosx

cosxcotx-2cosx=0
cosx(cotx-2)=0
cosx=0 cotx-2=0
x=cot-1 (2)
x=tan-1(1/2)

The key to solving this problem or problems like these is to move to one side, which is trig, and then factor.

2) cosx/secx+sinx/cscx

cosx/1/1/cosx + sinx/1/1/sinx = cos^2x+sin^2x = 1

Reflection 4/25

Okay, so this week was pretty good, but it went by too slow. Congrats to the softball team on going to state, and I'm sure everyone had a great time at prom and prom mania on Saturday night. The Trig exam is coming up, and hopefully all this trig will get back into my mind and I can understand it all. I'm going to review a couple of things in chapter 10 for this week's blog.

Sum and Difference Formula for Cosine

cos(alpha +- beta) = cos(alpha) cos(beta) -+ sin(alpha) sin(beta)

Find the exact value of cos 75
alpha = 45 beta = 30
cos(a + b) = cos(a)cos(b)-sin(a)sin(b)
cos 75 = (sqrt of 2/2)(sqrt of 3/2) - (sqrt of 2/2)(1/2)
cos 75 = (sqrt of 6 - sqrt of 2)/4


The only thing I'm gonna have trouble with is learning all of the formulas, and then maybe those problems where you have to edit the formula for example sin 4a = blank, or something like that...if there could be any example problems for this, that would be very much appreciated. Thanks. And also, i'm confused on when you have to find one angle, or two, or three...stuff like that.

Reflection 4/25

This had to be the funnest weekend everrr! The softball team had a game on saturday, which we won, and now we're going to STATE:). To add to that excitement, we also had prom on saturday night. It was nearly impossible for us to get ready in time after the softball game, but believe it or not, we made it. Prom mania was the GREATEST! There were sooo many prizes and it was sooo much fun! Anyway, back to math...right? So this trig exam? Its going to be hard or what? So i'm going to review for the trig exam, to help myself and others refresh on what is possibly on the exam. We have about one week left until we take the exam! Ouuu, yaaayy! Ha, but i'm pretty sure that everyone should do good on it. If you study and actually understand what we learned. But if you didn't pay attention in class, then you may have a problem. So anywayyy...
___________________________________________________________

Vertex:
a. (-b/2a, f(-b/2a)
b. complete the square to get into vertex form
y=(x+a)^2+b ((A & B ARE #s))
(-a,-b)

Focus:
1/4p= coefficient of x^2 then add p.
*if opens up, add to y-value from vertex.
*if down, subtract y-value from vertex.
*if opens right, add to x-value from vertex.
*if opens left, subtract.

Directrix is p units behind vertex
always x= or y=
so subtract p.


EXAMPLE:

x^2+1
-V:(0,1)
-focus:
1/4p=1
p=1/4
-directrix:
y=1-1/4
y=3/4

_________________________________________________________

Unit Circle:

90 degrees, (0,1), pi/2
360 degrees, (1,0), 2pi
180 degrees, (-1,0), pi
270 degrees, (0,-1), 3pi/2

sin = y/r
cos = x/r
tan = y/x
csc = r/y
sec = r/x
cot = x/y

EXAMPLES:

1.) sin 2pi = y/r = 0/1 = 0
2.) cos 180 degrees = x/r = -1/1 = -1
3.) sec 3pi/2 = r/x = 1/0 = undefined

__________________________________________________________

So the trig exam is only on chapters 7-11? If so, you can just give me some identity examples! I STILL don't understand how to do those. I guess i'll just have to sit down one night and memorize ALLLL of them. That is the only way i'll learn how to do those type of problems. But if anyone has any hints or examples, pleaseee?! :) HOPE EVERYONE HAD FUN AT PROM!

REFLECTION 4/25

For this week's blog I'll just review some things from Chapter 10. So here's a few examples:

Ex. 1.) If sinx=12/13 Find tan x/2
*Okay for this problem they're asking you to find tan x/2, and that has a formula:
sinx/(1+cos x) So you're going to use that formula to find the answer
*So first you have to draw a triangle from what you're given (sinx=12/13)
(So the height of the triangle is 12, the hypotenuse is 13, and the base is 5)
*Now you can plug into the formula:
tan x/2 = sin x/(1+cos x)
=12/13 / 1+(5/13)
= 12/13 / 18/13 then sandwich it
=156/234 = 2/3
*So tan x/2 is 2/3

Ex. 2.) Simplify tan75-tan30/(1+tan75tan30)
*When you first look at this problem you should notice that it is a formula
*It's a formula for tan(a-b) **a=alpha;b=beta
*So all you do is plug the two numbers you're given into tan(a-b) where the bigger number is alpha and the other number is beta (So a=75 and b=30)
*So you get tan(75-30)
= tan 45
And they're asking you to simplify, so that means your answer should be a number
*tan 45 is 1, so 1 is your answer

Ex. 3.) Simplify cos(60 + y) + cos(60 - y)
*For this problem you're going to have to use the cosine sum and difference formulas and you'll have to work the two parts of the problem separately
*So first you can solve cos(60 + y) Plugging that into the formula you get:
(cos60cosy - sin60siny)
^^And that simplifies to give you (1/2)cosy - sqrt of 3/2siny
*Now you can solve cos(60 - y)
So you get: (cos60cosy-sin60siny)
^And that simplifies to (1/2)cosy + sqrt of 3/2siny
Now you add everything together:
(1/2cosy - sqrt of 3/2siny + 1/2cosy + sqrt of 3/2siny
sqrt of 3/2's cancel out so you're left with 1/2cosy + 1/2cosy
And that equals 1cosy => cos y is your answer

Ex. 4.) If tan x = 1/2 and tan y = 2/3 Find tan(x+y)
*they're asking for tan(x+y) so all you have to do is plug the numbers into the tangent sum formula:
tna(a+b) = tanz+tanb/(1-tanatanb)
*so you get 1/2 + 2/3 / 1-(1/2)(2/3)
= 7/6 / 1-1/3
=7/6 / 2/3 sandwich it and you get 21/12
*So your simplified answer is 7/4