This week was great! :) We barely went to school..that's how I like it. Anyways, since we didn't really learn anything new this week i'll just explain somethings in chapter 10.
Problems like these will be on the chapter test:
If tan a = 2 and tan B = -1/3 then find tan (a+B)
*First you figure out which tan formula to use.
tan (a+B) = tan a + tan B/1-tan a tan B
tan (a-B) = tan a - tan B/1+tan a tan B
*Once you figure out which one to use plug in the numbers they give you into the formula. (Make sure you replace the whole tan a, or tan b. Not just the letter)
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
Find sin 150
*The first thing you notice is that 150 is not on your trig chart.
*First you have to find out which quadrant 150 is in to see if sin is negative or positive.
(keep in mind that sine relates to the y axis)
150 is in the 2nd quadrant, making sin positive because the y axis is positive.
*Now to find the reference angle you subtract 150 from 180 and you get 30.
So you get sin 30. The exact value of sin 30 is 1/2
----------------------------------------
I don't fully understand the half angle stuff..or when you have to convert stuff to degrees. I'm not always sure if you have to convert it to degrees or if you could just work the problem leaving it the way it is.
Saturday, January 16, 2010
Friday, January 15, 2010
Reflection #23
alright for this week we learned all about chapter 10 and learned how to use formulas to help us condense, expand, and solve problems with trig functions.
formulas:
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
something that i understood the most was section 2 here's some examples:
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
for something that i didn't understand from this chapter was for example was in section 4 when you use the formulas as identities. i don't understand how you use the unit circle or the four quadrants to determine the degrees like in the homework on page 389 numbers 12 & 13.
formulas:
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
something that i understood the most was section 2 here's some examples:
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
for something that i didn't understand from this chapter was for example was in section 4 when you use the formulas as identities. i don't understand how you use the unit circle or the four quadrants to determine the degrees like in the homework on page 389 numbers 12 & 13.
Thursday, January 14, 2010
Reflection 22 (really late, i forgot)
So all this formula stuff has been really tough, an not studyin as much as i need to is not helpin at all. I'ma come try to explain somethin right quick but I can't promise i'm exactly right cuz i forgot ma notebook in ma locker today.
So sin(a + b)=sina sinb - cosa cosb
when do you use this? say you have to solve sin 105
Break it down by usin the formula by sayin, what plus what equals 105 (hintboth values will usually be from the trig chart) so in this cse it would be 60 and 45.
So now let's plug into the formula:
sin(60 + 45) = sin60 sin45 - cos60 cos45
use the trig chart to take it a step farther:
sin(60 + 45) = (square root of 3/2)(square root of 2/2) - (1/2)(square root of 2/2)
which, simplified equals: square root of 6 - square root of 2/4
That's it.
**Hint, if you get a low value to find, like sin15, don't forget that you can flip the formula:
sin(a - b) sina sinb + cosa cosb
So in this case you would use sin(60 - 45) and solve it like you solved the first example.
__________________________________________________________
I don't understand how to fin tan15 using the half angle formula. Help?
So sin(a + b)=sina sinb - cosa cosb
when do you use this? say you have to solve sin 105
Break it down by usin the formula by sayin, what plus what equals 105 (hintboth values will usually be from the trig chart) so in this cse it would be 60 and 45.
So now let's plug into the formula:
sin(60 + 45) = sin60 sin45 - cos60 cos45
use the trig chart to take it a step farther:
sin(60 + 45) = (square root of 3/2)(square root of 2/2) - (1/2)(square root of 2/2)
which, simplified equals: square root of 6 - square root of 2/4
That's it.
**Hint, if you get a low value to find, like sin15, don't forget that you can flip the formula:
sin(a - b) sina sinb + cosa cosb
So in this case you would use sin(60 - 45) and solve it like you solved the first example.
__________________________________________________________
I don't understand how to fin tan15 using the half angle formula. Help?
Wednesday, January 13, 2010
Monday, January 11, 2010
reflection 22
Okay, so this week wasnt that bad. I didnt really understand everything. it confuses me a lot. Here's what we learned in general as far as formulas:
Sum and Difference formulas for Cosine and Sine:
cos(A+-B) = cos A cos B -+ sin A sin B
sin(A+-B) = sin A cos B +- cos A sin B
Rewriting sum and difference as a product:
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)
Sum formula for Tangent:
tan(A+B)= tan A+ tan B/ 1-tanAtanB
Difference formula for tangent:
tan(A-B) = tan A-tan B/ 1+tanAtanB
and for what i dont know, I dont understand when it asks to find the exact value of something. it just confuses me. i forgot like all of my algebra so its not nice. haha.
Sum and Difference formulas for Cosine and Sine:
cos(A+-B) = cos A cos B -+ sin A sin B
sin(A+-B) = sin A cos B +- cos A sin B
Rewriting sum and difference as a product:
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)
Sum formula for Tangent:
tan(A+B)= tan A+ tan B/ 1-tanAtanB
Difference formula for tangent:
tan(A-B) = tan A-tan B/ 1+tanAtanB
and for what i dont know, I dont understand when it asks to find the exact value of something. it just confuses me. i forgot like all of my algebra so its not nice. haha.
reflection 22
This week in math was alright, but the week flew by wich was good. We been learning alot of formuals and the week is pretty easy if u know the formulas its just the fact of having to remember them all. Here are some of the many formuals:
DOUBLE AND HALF-ANGLE FORMULAS
sin2(alpha)=2sin(alpha) cos(alpha)
cos2(alpha)=cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan2(alpha)=2tan(alpha)/1-tan^2(alpha)
sin(alpha)/2= +or- sqrt of (1-cos(alpha)/2)
cos(alpha)/2=+or- sqrt of (1+cos(alpha)/2)
tan(alpha)/2=+or- sqrt of (1-cos(alpha)/1+cos(alpha)=sin(alpha)/1+cos(alpha)=1-cos(alpha)/sin(alpha)
Sum and Difference Formulas for Cosine and Sine
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta)
Difference Formula for Tangent:
tan(alpha-beta)=tan(alpha)-tan(beta)/1+tan(alpha) tan(beta)
Sum Formula for Tangent:
tan(alpha+beta)=tan(alpha)+tan(beta)/1-tan(alpha) tan(beta)
Rewriting a Sum or Difference as a Product
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)
Yea all of these wondeful formulas. If i could just remember all of them this may be the easiest lesson, and NOTE TO ALL!!!!!!! STUDY YOUR TRIG CHART!!!!!!!
and does anyone remember what pg number the hw was on for friday?
DOUBLE AND HALF-ANGLE FORMULAS
sin2(alpha)=2sin(alpha) cos(alpha)
cos2(alpha)=cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan2(alpha)=2tan(alpha)/1-tan^2(alpha)
sin(alpha)/2= +or- sqrt of (1-cos(alpha)/2)
cos(alpha)/2=+or- sqrt of (1+cos(alpha)/2)
tan(alpha)/2=+or- sqrt of (1-cos(alpha)/1+cos(alpha)=sin(alpha)/1+cos(alpha)=1-cos(alpha)/sin(alpha)
Sum and Difference Formulas for Cosine and Sine
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta)
Difference Formula for Tangent:
tan(alpha-beta)=tan(alpha)-tan(beta)/1+tan(alpha) tan(beta)
Sum Formula for Tangent:
tan(alpha+beta)=tan(alpha)+tan(beta)/1-tan(alpha) tan(beta)
Rewriting a Sum or Difference as a Product
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)
Yea all of these wondeful formulas. If i could just remember all of them this may be the easiest lesson, and NOTE TO ALL!!!!!!! STUDY YOUR TRIG CHART!!!!!!!
and does anyone remember what pg number the hw was on for friday?
Sunday, January 10, 2010
Reflection 23
Here are the identities that we learned this week that we had to memorize along with the trig chart.
Difference Formula for Tangent:
tan(alpha-beta)=tan(alpha)-tan(beta)/1+tan(alpha) tan(beta)
Sum Formula for Tangent:
tan(alpha+beta)=tan(alpha)+tan(beta)/1-tan(alpha) tan(beta)
Sum and Difference Formulas for Cosine and Sine:
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta
)
DOUBLE AND HALF-ANGLE FORMULAS:
sin2(alpha)=2sin(alpha) cos(alpha)
cos2(alpha)=cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan2(alpha)=2tan(alpha)/1-tan^2(alpha)
sin(alpha)/2= +or- sqrt of (1-cos(alpha)/2)
cos(alpha)/2=+or- sqrt of (1+cos(alpha)/2)
tan(alpha)/2=+or- sqrt of (1-cos(alpha)/1+cos(alpha)=sin(alpha)/1+cos(alpha)=1-cos(alpha)/sin(alpha)
Im confused in trying to solve degrees that are larger than the trig chart goes. Anybody can help?
Difference Formula for Tangent:
tan(alpha-beta)=tan(alpha)-tan(beta)/1+tan(alpha) tan(beta)
Sum Formula for Tangent:
tan(alpha+beta)=tan(alpha)+tan(beta)/1-tan(alpha) tan(beta)
Sum and Difference Formulas for Cosine and Sine:
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta
)
DOUBLE AND HALF-ANGLE FORMULAS:
sin2(alpha)=2sin(alpha) cos(alpha)
cos2(alpha)=cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan2(alpha)=2tan(alpha)/1-tan^2(alpha)
sin(alpha)/2= +or- sqrt of (1-cos(alpha)/2)
cos(alpha)/2=+or- sqrt of (1+cos(alpha)/2)
tan(alpha)/2=+or- sqrt of (1-cos(alpha)/1+cos(alpha)=sin(alpha)/1+cos(alpha)=1-cos(alpha)/sin(alpha)
Im confused in trying to solve degrees that are larger than the trig chart goes. Anybody can help?
Reflection #21
Okayy, we are now on chapter ten and it is pretty easy if you learn all of the formulas. You should probably memorize all of the formulas as soon as she gives them to us, so you won't fail the quizzes.
_____________________________________________________________________
Sum and Difference Formulas for Cosine and Sine:
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta)
EXAMPLE:
Find the exact value of sin15degrees.
alpha=45degrees and beta=30degrees
sin(45-30)=sin(45)cos(30)-cos(45)sin(30)
sin(15)=(square root of 2/2)(square root of 3/2)-(square root of 2/2)(1/2)
sin(15)= square root of 6 - square root of 2 all over 4
_____________________________________________________________________
Sum formula for Tangent:
tan(A+B)= tan A+ tan B/ 1-tanAtanB
Difference formula for tangent:
tan(A-B) = tan A-tan B/ 1+tanAtanB
EXAMPLE:
Find the exact value of tan15+tan30/1-tan15+tan30
=tan(15+30)
=tan(45)=1
**use the trig chart to find the exact value.
_________________________________________________________________
I pretty much get how to do all of these problems, i just need to memorize all of them for the test. If anyone would like to give me some examples of these problems, please do so. I need all the help i can get. Please and Thank You!
_____________________________________________________________________
Sum and Difference Formulas for Cosine and Sine:
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta)
EXAMPLE:
Find the exact value of sin15degrees.
alpha=45degrees and beta=30degrees
sin(45-30)=sin(45)cos(30)-cos(45)sin(30)
sin(15)=(square root of 2/2)(square root of 3/2)-(square root of 2/2)(1/2)
sin(15)= square root of 6 - square root of 2 all over 4
_____________________________________________________________________
Sum formula for Tangent:
tan(A+B)= tan A+ tan B/ 1-tanAtanB
Difference formula for tangent:
tan(A-B) = tan A-tan B/ 1+tanAtanB
EXAMPLE:
Find the exact value of tan15+tan30/1-tan15+tan30
=tan(15+30)
=tan(45)=1
**use the trig chart to find the exact value.
_________________________________________________________________
I pretty much get how to do all of these problems, i just need to memorize all of them for the test. If anyone would like to give me some examples of these problems, please do so. I need all the help i can get. Please and Thank You!
REFLECTION 21
Hey guys, no school tomorrow!! woop woop! haha, so on to the math!
im gonna do the formulas, which im sure everyone else is doing! haha.
DOUBLE AND HALF-ANGLE FORMULAS:
sin2(alpha)=2sin(alpha) cos(alpha)
cos2(alpha)=cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan2(alpha)=2tan(alpha)/1-tan^2(alpha)
sin(alpha)/2= +or- sqrt of (1-cos(alpha)/2)
cos(alpha)/2=+or- sqrt of (1+cos(alpha)/2)
tan(alpha)/2=+or- sqrt of (1-cos(alpha)/1+cos(alpha)=sin(alpha)/1+cos(alpha)=1-cos(alpha)/sin(alpha)
Difference Formula for Tangent:
tan(alpha-beta)=tan(alpha)-tan(beta)/1+tan(alpha) tan(beta)
Sum Formula for Tangent:
tan(alpha+beta)=tan(alpha)+tan(beta)/1-tan(alpha) tan(beta)
Sum and Difference Formulas for Cosine and Sine:
cos(alpha+or-beta)=cos(alpha) cos(beta)-or+sin(alpha) sin(beta)
sin(alpha+or-beta)=sin(alpha) cos(beta)+or-cos(alpha) sin(beta)
-------------------------------------------------------------------------------
So, as for what I don't know:
I basically know the formulas, I just don't know how to use them; like I don't know what to put in alpha and beta and which one to put it in....
I need help because I've been failing every quiz so far and I feel really stupid!
PLEASEEE HELPPPP!!!!!!!!
Reflection 21!
Mkay so i pretty much understood everything this week. I don't know about the second quiz, but I did good on the first one! :)
I'll just use this reflection to go over the formulas we learned this week, the more i write them the more i'll remember.... hopefully.
Sum and Difference formulas for Cosine and Sine:
cos(A+-B) = cos A cos B -+ sin A sin B
sin(A+-B) = sin A cos B +- cos A sin B
Rewriting sum and difference as a product:
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)
Sum formula for Tangent:
tan(A+B)= tan A+ tan B/ 1-tanAtanB
Difference formula for tangent:
tan(A-B) = tan A-tan B/ 1+tanAtanB
-----------------
S for what i don't understand,
I'm still learning how to use the half-angle formula in an equation like:
Find the exact value for sin115degrees.
help?
thanks!
I'll just use this reflection to go over the formulas we learned this week, the more i write them the more i'll remember.... hopefully.
Sum and Difference formulas for Cosine and Sine:
cos(A+-B) = cos A cos B -+ sin A sin B
sin(A+-B) = sin A cos B +- cos A sin B
Rewriting sum and difference as a product:
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)
Sum formula for Tangent:
tan(A+B)= tan A+ tan B/ 1-tanAtanB
Difference formula for tangent:
tan(A-B) = tan A-tan B/ 1+tanAtanB
-----------------
S for what i don't understand,
I'm still learning how to use the half-angle formula in an equation like:
Find the exact value for sin115degrees.
help?
thanks!
Reflection
Well this week wasn't too bad. Just lots of identities to know. Also we have to know the trig chart since we can't use our calculators. Here is an example from what we have learned this week:
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
All you really have to do is figure out what formula you are using which is obviously the subraction tangent formula. And again don't forget about your trig chart.
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
All you really have to do is figure out what formula you are using which is obviously the subraction tangent formula. And again don't forget about your trig chart.
reflection 21
This week went by too slow for me. But no school tomorrow or wednesday!:) We learned a lot of things in chapter 10 this week. There are so many formulas that you have to know for every little thing. Heres some stuff from 10-1.
Find the exact value of sin15 degrees.
a=45 degrees B=30 degrees
sin(a-B)-sinacosB-cosasinB
sin(45-30)=sin45cos30-cos45sin30
sin 15= (squareroot of 2/2)(squareroot of 3/2) - (squareroot of 2/2)(1/2)
sin 15= squareroot of 6/4 - squareroot of 2/4
sin15= squareroot of 6 - squareroot of 2/4
...this came from the sum and differece formulas for sin and cos.
Heres some stuff from 10-2.
Find the exact value of tan15+tan30/1-tan15tan30.
=tan(15+30)
=tan(45)
=1
...the sum formula for tangent was used in that one.
Find the exact value of sin15 degrees.
a=45 degrees B=30 degrees
sin(a-B)-sinacosB-cosasinB
sin(45-30)=sin45cos30-cos45sin30
sin 15= (squareroot of 2/2)(squareroot of 3/2) - (squareroot of 2/2)(1/2)
sin 15= squareroot of 6/4 - squareroot of 2/4
sin15= squareroot of 6 - squareroot of 2/4
...this came from the sum and differece formulas for sin and cos.
Heres some stuff from 10-2.
Find the exact value of tan15+tan30/1-tan15tan30.
=tan(15+30)
=tan(45)
=1
...the sum formula for tangent was used in that one.
Reflection 21
Well, this week went by extremely slow, but im finally glad it's over and im happy to say that there is NO SCHOOL TOMORROW and Wednesday! So we learned a lot in advanced math this week, and we mostly learned formulas and how to apply them. Here is a list of the formulas we learned.
Sum and Difference formulas for cosine and sine
cos (alpha + or - beta) = cos(alpha)cos(beta) - or + sin(alpha)sin(beta)
sin (alpha + or - beta) = sin(alpha)cos(beta) + or - cos(alpha)sin(beta)
Rewriting a Sum or Difference as a Product
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)
Half angle and double angle formulas
sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos(alpha)=
1-cos(alpha)/sin(alpha)
Now the only thing i don't get is when you have to solve the equations and you have something like sinx=cosx...help?
Sum and Difference formulas for cosine and sine
cos (alpha + or - beta) = cos(alpha)cos(beta) - or + sin(alpha)sin(beta)
sin (alpha + or - beta) = sin(alpha)cos(beta) + or - cos(alpha)sin(beta)
Rewriting a Sum or Difference as a Product
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)
Half angle and double angle formulas
sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos(alpha)=
1-cos(alpha)/sin(alpha)
Now the only thing i don't get is when you have to solve the equations and you have something like sinx=cosx...help?
Reflection 21.
Well the first week back at school wasn't too bad in math. I understand everything we learned this week it was basic simple math. You were just plugging into formulas pretty much, and knowing when to add numbers from the trig chart together. For this chapter you have to know your trig chart or you will fail. *On some problems when you get to the end of a problem and it gives you an answer like tan 45 then you go to the trig chart and find the answer (which would be 1).
These are all the formulas:
Cos(α +/- β)=cos α cos β -/+ sin α sin β
Sin(α +/- β)=sin α cos β -/+ cos α sin β
*Here's an example of when you can use these formulas.
Ex: Find the exact value of sin 15degrees
*Note that when asked for exact value it means you will use your trig chart.
-From here you think what two numbers from the trig chart can either add or subtract to give you the number it wants you to find. (45-30)
-Since you would do 45-30 and it's sin you would look for the formula that goes with that.
Then plug the numbers into alpha and beta, then plug the numbers in from the trig chart.
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
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
*^We didn't do anything with these formulas yet, so you don't really use thse for the problems.
Tan (α + β)=tan α + tan β/1-tan α tan β
Tan (α - β)=tan α - tan β/1+tan α tan β
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
*On this you are just replacing the tan alphas and betas with the number. Be sure you replace the whole thing (tan included) not just the alpha or beta.
---------------------------------------------
Some of them can be confusing sometimes I just have to remember all the formulas and know what steps to do where.
These are all the formulas:
Cos(α +/- β)=cos α cos β -/+ sin α sin β
Sin(α +/- β)=sin α cos β -/+ cos α sin β
*Here's an example of when you can use these formulas.
Ex: Find the exact value of sin 15degrees
*Note that when asked for exact value it means you will use your trig chart.
-From here you think what two numbers from the trig chart can either add or subtract to give you the number it wants you to find. (45-30)
-Since you would do 45-30 and it's sin you would look for the formula that goes with that.
Then plug the numbers into alpha and beta, then plug the numbers in from the trig chart.
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
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
*^We didn't do anything with these formulas yet, so you don't really use thse for the problems.
Tan (α + β)=tan α + tan β/1-tan α tan β
Tan (α - β)=tan α - tan β/1+tan α tan β
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
*On this you are just replacing the tan alphas and betas with the number. Be sure you replace the whole thing (tan included) not just the alpha or beta.
---------------------------------------------
Some of them can be confusing sometimes I just have to remember all the formulas and know what steps to do where.
Reflection #21
Okay, chapter 10!
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, what I don't understand is...problems like number 21-26 on p. 373. I'm pretty sure those are the only ones I don't know how to do, but I'm not even sure how to start. I'm probably making it out to be a lot more complicated than it actually is, but if any would care to explain, that would be great! ^^
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, what I don't understand is...problems like number 21-26 on p. 373. I'm pretty sure those are the only ones I don't know how to do, but I'm not even sure how to start. I'm probably making it out to be a lot more complicated than it actually is, but if any would care to explain, that would be great! ^^
REFLECTION #21
Okay well this week we started Chapter 10 which consists of A LOT of formulas! The whole chapter basically revolves around every formula we learned. I had trouble with what we were taught on Monday, but as we kept doing example problems and whatnot, I understood it much better. Anyway, Monday we learned the sum and difference formulas for sine and cosine. Those formulas are used typically to find the exact value of an angle that is not found on the trig chart. They're used for other things too. (*and angles on the trig chart have to add or subtract to get that number..) I thought this was pretty easy so I'll give an example and explain one of these types of problems.
Ex 1.) Find the exact value of cos 105.
*Okay the first thing you notice is that cos 105 is not on your trig chart.
*Then you have to think if there are any numbers on the trig chart that add to give you 105.
*The angles 60 and 45 degrees add to get 105 degrees, so you can use that in your equation.
*So you're going to use the cosine sum formula. (*remember when using the cosine formula, the signs are opposite) (**and also to make things clearer, I'll make a = alpha and b = beta)
*so the equation for cosine sum is: cos(a + b) = cosacosb - sinasinb
*Now you just plug in your numbers: 60 and 45
*So you get: cos60cos45 - sin60sin45
*Then you can use your trig chart to simplify those functions and you get this:
= (1/2)(square root of 2/2) - (square root of 3/2)(square root of 2/2)
*Then you multiply those together and subtract them and your final answer is:
square root of 2 - square root of 6 / 4
*Then on Tuesday we learned the tangent formulas. Those were pretty easy too.
Here's an example of one of those:
Ex. 2.) tan a = 1/4 tan b = 3/5
Find tan (a + b) (*again, a is alpha and b is beta)
*So for this problem all it's telling you to do is use the tan (a + b) formula which is this:
tan (a + b) = tan a + tan b/1 - tan a tan b
*All you have to do is completely substitute the numbers in place of the functions.
* So you get this when you use the formula: 1/4 + 3/5 / 1 - (1/4)(3/5)
*Then you add the fractions on top and get 17/20.
*And multiply the fractions on the bottom and then subtract from 1 and you get 17/20
*Then you sandwich it!!!!!!!
*And you get 340/340 which equals 1
*Then on Wednesday we learned the double angle and half angle formulas. Those are pretty simple because all you have to do is memorize the formulas and you should be good.
Here's an example problem:
Ex. 3.) sin 2A / 1 - cos 2A = cot A
*For this problem it tells you that the left side of the problem equals cot A. All you have to do is show how it equals that. Basically what you want to do is expand then condense the left side of the problem to make it look like the right side.
*So the first thing you can do is use your identities to change sin 2A to 2sinAcosA
*And at the bottom of the fraction you can change cos 2A to 1 + 2sin^2A
*So your fraction now looks like this: 2sinAcosA/1-(1+2sin^2A)
*Then the 1's cancel out on the bottom of the fraction so you get this: 2sinAcosA/2sin^2A
*Then you can cancel out a 2sin since there's one on the top and bottom of the fraction
*And you get cosA/sinA
*Using your identities that reduces to cot A
**Okay now for what I don't understand, which isn't too much this week surprisingly. Well, I'll be specific with this. I'm still having a little trouble with problems like numbers 33, 34, 37 and 38 on page 384. I'm not sure what identities to use to solve them. So if anyone would like to help me out with that, that would be greatly appreciated (:
Ex 1.) Find the exact value of cos 105.
*Okay the first thing you notice is that cos 105 is not on your trig chart.
*Then you have to think if there are any numbers on the trig chart that add to give you 105.
*The angles 60 and 45 degrees add to get 105 degrees, so you can use that in your equation.
*So you're going to use the cosine sum formula. (*remember when using the cosine formula, the signs are opposite) (**and also to make things clearer, I'll make a = alpha and b = beta)
*so the equation for cosine sum is: cos(a + b) = cosacosb - sinasinb
*Now you just plug in your numbers: 60 and 45
*So you get: cos60cos45 - sin60sin45
*Then you can use your trig chart to simplify those functions and you get this:
= (1/2)(square root of 2/2) - (square root of 3/2)(square root of 2/2)
*Then you multiply those together and subtract them and your final answer is:
square root of 2 - square root of 6 / 4
*Then on Tuesday we learned the tangent formulas. Those were pretty easy too.
Here's an example of one of those:
Ex. 2.) tan a = 1/4 tan b = 3/5
Find tan (a + b) (*again, a is alpha and b is beta)
*So for this problem all it's telling you to do is use the tan (a + b) formula which is this:
tan (a + b) = tan a + tan b/1 - tan a tan b
*All you have to do is completely substitute the numbers in place of the functions.
* So you get this when you use the formula: 1/4 + 3/5 / 1 - (1/4)(3/5)
*Then you add the fractions on top and get 17/20.
*And multiply the fractions on the bottom and then subtract from 1 and you get 17/20
*Then you sandwich it!!!!!!!
*And you get 340/340 which equals 1
*Then on Wednesday we learned the double angle and half angle formulas. Those are pretty simple because all you have to do is memorize the formulas and you should be good.
Here's an example problem:
Ex. 3.) sin 2A / 1 - cos 2A = cot A
*For this problem it tells you that the left side of the problem equals cot A. All you have to do is show how it equals that. Basically what you want to do is expand then condense the left side of the problem to make it look like the right side.
*So the first thing you can do is use your identities to change sin 2A to 2sinAcosA
*And at the bottom of the fraction you can change cos 2A to 1 + 2sin^2A
*So your fraction now looks like this: 2sinAcosA/1-(1+2sin^2A)
*Then the 1's cancel out on the bottom of the fraction so you get this: 2sinAcosA/2sin^2A
*Then you can cancel out a 2sin since there's one on the top and bottom of the fraction
*And you get cosA/sinA
*Using your identities that reduces to cot A
**Okay now for what I don't understand, which isn't too much this week surprisingly. Well, I'll be specific with this. I'm still having a little trouble with problems like numbers 33, 34, 37 and 38 on page 384. I'm not sure what identities to use to solve them. So if anyone would like to help me out with that, that would be greatly appreciated (:
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