WEBVTT
Kind: captions
Language: en
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so first
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most of you
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trigonometry, make sure everyone remembers their trigonometry
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the first thing is
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sin, cosine ad tangents
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so what is sine of an angle
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the sin of angles and soh cah toa
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you have a right triangle
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you have the angle x
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you have 3 sides, you have side opposite
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you have the side adjacent
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and you have the hypothenuse
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and the sine of the angle
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is the ratio of the opposite sides
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and hypothenuse, remember
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al the trigonometry stuff should have taught you this
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the ratio of he triangle stays the same even if the size of the triangle changes
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if you have a right triangle and a particular angle, the ratio never changes
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we give names for the ratios to make it easier o work with
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they are called the sinx the ratio
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of this opposite side and the hypothenuse
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and then cosine of x
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is the adjacent side
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over the hypothenuse, and the tangent
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is the opposite
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over the adjacent
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and you have to remember all that
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sohcahtoa
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thats up on the instagram
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you guys all know this stuff right?
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so if you didnt know it the last time around, this is a good chance to do it
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cause were gonna test you on this stuff again
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in the certain magic triangles that you work with
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one of the kind we work with is the 45, 45 90
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and the 45, 45, 90 triangle
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comes from cutting the square
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in half on the diagonal
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if you cut a square in half on the diagonal, these two sides are still equal, still a square
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and that side
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is x radical 2
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the sine of that angle, sorry that angle is always 45 degrees
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and that angle is 45 degrees
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you want to find the sign of 45 degrees
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its just the ratio of the opposite side and the hypothenuse
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now if you cancle the x's
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is 1/square root of 2
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and then if you rationalize that
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multiply top and bottome by 2
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it would also equal 2/2
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this is stuff you want to make surely in your brain
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because when you do a lot of this
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in the next month, plus those lucky enough to take calculus
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you see a lot of trignometry in calculus
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i had a problem today, to find which sine and cosine intersect
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sine and cosine are the same
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sine and cosine are the same, theyre 45 degree angkes
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youre suppose to remember that all the way till MAT126
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and of course
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so when youre 50 years old, really old
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and youre walking down the street and someone said whats the cosine of 45 degrees
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dont get a ride before you do
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i who will still be a live at that point, would say thank you very much
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maybe well i hope so
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im still oyung exactly
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cosine of 45 degrees
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is also
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x/x radical 2
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why are they the same?
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because these 2 sides are the same
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and thats also the square root of 2
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radical 2/2
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the tangent of x, x is 45
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well the two sides are the same, the opposite and the adjacent
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he tangent of 45 is 1
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thats how youre gonna master the chart
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you want to make sure you know these
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so far so good? cause im gonna cover this up in a minute
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now were gonna do 30, 60, 90 triagnle
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now suppose you take and equilateral triangle
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and you cut it in half, well and equilateral triangle
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when you cut it in half you divide the base
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in half. the base
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and this side is the same as a equilateral triangle
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this is x and this is 2x
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you can write the hypothenuse and this comes out
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x square root of 3
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thats why we know alll the time
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so if i want to find the sine of 30 degrees
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you get x
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x to the 2x
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is 1/2
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and the cosine of 30 degrees
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is x radical 3
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over 2x
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which is radical 3/2
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and the tangent of 30 degrees
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is x over x times the square root of 3
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1/square root of 3 or if you want to rationalize
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radical 3 over 3
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make sure
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that youre comfortable with
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radical 3 over 3 and 1/ square root of 3
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you know both are the same thing
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just like radical 3/2 and 1 over square root of 2
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radical x over x
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is the same as 1/ square root of x
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radical 9 well radical 10 over 10 is the same as 1/ radical 10
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they are all the same
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now if i want to do 60 degrees
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well the sine of 60 degrees, thats a 60 degree angle
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is the opposite over the hypothenuse, thats the smae
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as the cosine of 30 degrees
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and the cosine of 60 degrees
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is the same as sine 30 degrees
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the tangent of 60 degrees
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is x radical 3/x
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just comes out radical 3
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you got all this down?
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good cause im gonna make a little chart
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make sure you remember
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ill put the chart right heree
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so you can kind of see it
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so when i showed you before
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you can make this into a nice little chart
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where you can make this sin, cosine and tangent
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30 45 and 60 degrees on top
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and then you can just fill in
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notice the sin of 30 degrees is 1/2
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sin of 45 degrees
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radical 2/3 over here
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bit wait it was up there and now its down here
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alright so
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remember when you do the sine, everything is over 2
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1, 2,3
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and then for cosine everything is over 2 again
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you can do
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3
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2 and 1
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thats a very good way to memorize
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the sine and cosines
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the tangent is found by taking the sine
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and dividing it by the cosine
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so sine over cosine is 1
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cosine/ sine is square root of 3
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alright that was our basic
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right triangle trick
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now theres other stuff you should know about right triangle ratio
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see that right there?
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so were gonna do the pythagorean theorem
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so then for this triangle, a square plus b squared
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is c squared
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now what if i take everything and divide it by c squared
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well now i got a squared over c square
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or a over c
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squared
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b/c
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squared
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is 1
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what is a/c, a/c is the sine of x
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b/c is the cosine of x
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the sine squared plus the cosine squared
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always equals 1 as long as they are the same angle
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so thats another thing you want to make sure you remember
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were gonna do trig identities in a week, so you got to make sure you remember these
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another thing is
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notice the tangent of x
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well thats a/b
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but if you took a and divide it by c
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and b/c
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you still have a.b
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and a/c is the sin
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and b/c is the cosine
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so the tagent of any angle
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is the sine of the angle divided by the cosine of the angle
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remember all this so far?
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hope so
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for those of you who didnt do well on minimum competent on the first midterm this is a good time to remember this stuff
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thats another identity you want to make sure you know
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next thing we did
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we call this trig stuff
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and we used this to find angles
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other then
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your 90 degrees
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so how do you do that?
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you draw a circle with the radius 1
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one is also known as
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so this is our unit circle
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say if you want to find, sin and cosine what you do
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you can imagine radius
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is coming around the circle
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if you look at the angle and the radius
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the x axis
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this distance, oh lets call that theta
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this distance is x this distance is y
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that gives us the coordinates x,y
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but see this is 1
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the sin of theta is y/1
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and the cosine of theta is x/1
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so the coordinates is also just
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cosine theta, sin theta
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what you can do is imagine going aroun in a circle
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and where ever you stop
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the coordinates of that point
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are the cosine of theta around where you stop
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and the sine of the theta where you stop
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thats why you call it being a circle
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so you start here
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go around to some point and stop here
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and these coordinates
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are the sine
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that
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and the cosine of that
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thats why we use that
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that was this trick that
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that enable people to do sin cosine of angles
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other then
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between 0 and 90 degrees
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but they said
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i cant have a right triangle of an angle of 260 degrees
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you have to find another way to find the sin cosine of that
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so thats means we have to be able to shift quadrants
00:15:34.660 --> 00:15:36.420
and to find, sin and cosine
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so if you want to find the sin cosine of somethng you take the quadrant
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and you just figure out what the triangle would be
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and relate it back to the first one
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and you look and notice the coordinates change there signs, so in the first quadrant
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x and y are both positive so the sin and cosine are both positive
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in the second quadrant
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the x coordinate is negative
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the y coordinate stays positive
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so the sign will stay positive
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but the cosine will be negative
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in the third quadrant
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x and y
00:16:09.920 --> 00:16:11.760
are both negative down here
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so both sin and cosine will be negative
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and in the 4th quadranr
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dow here
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x is positive and y is negative so
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cosine will be positive and y is negative
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so alittle thing to help us remember that
00:16:33.960 --> 00:16:37.240
that reminds us whats positive and what negative
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so the first quadrant, so if you have an angle in the first quadrant
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so between 0 and 90 degrees
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call this 0 degrees and this 90 degrees
00:16:45.660 --> 00:16:48.220
all of the trig functions are positive
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if you go to the second quadrant
00:16:52.900 --> 00:16:54.660
between 90 and 180 degrees
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now the sine stays positive, the sign stays positive
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because the y coordinate is
00:16:59.960 --> 00:17:01.000
still positive
00:17:01.000 --> 00:17:04.160
the x coordinate is now negative because its going left
00:17:05.340 --> 00:17:06.940
so only sine is positive
00:17:09.000 --> 00:17:11.400
if you go down from 180 to 270 degrees
00:17:13.380 --> 00:17:15.620
now the tangent becomes positive
00:17:15.620 --> 00:17:17.540
but the sin and cosine become negative
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why is the tangent positive?
00:17:19.020 --> 00:17:20.220
when x is negative
00:17:20.700 --> 00:17:21.580
y is negative
00:17:22.040 --> 00:17:23.800
the ratio you get positive
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and then you go to the last quadrant
00:17:30.000 --> 00:17:31.120
fourth quadrant
00:17:32.700 --> 00:17:36.140
the cosine is positive, why is the cosine positive?
00:17:36.140 --> 00:17:40.140
because the x coordinates are positive and the y coordinates are negative
00:17:41.180 --> 00:17:42.300
x is cosine theta
00:17:42.700 --> 00:17:43.900
and y is sine theta
00:17:45.660 --> 00:17:46.620
so far so good?
00:17:48.780 --> 00:17:51.500
okay everybody remember all this stuff?
00:17:52.660 --> 00:17:53.160
good
00:17:56.800 --> 00:17:57.440
so you all
00:17:58.960 --> 00:18:01.040
should be able to find a problem
00:18:04.400 --> 00:18:08.640
were gonna practice right now before we move onto the good stuff
00:18:35.320 --> 00:18:37.320
were gonna do those 5 right now
00:18:48.440 --> 00:18:52.840
you all should be able to do these, notice i havent done radians yet
00:18:52.840 --> 00:18:54.740
got to convert it into radains
00:18:57.160 --> 00:18:58.760
so if you want to do these
00:18:58.760 --> 00:19:01.140
each of these is the exact same technique
00:19:01.280 --> 00:19:04.560
so you want to make sure you are really good at this
00:19:05.260 --> 00:19:09.260
because this is were people fell down in minimum competence
00:19:09.260 --> 00:19:10.940
with these kind of problems
00:19:11.920 --> 00:19:14.480
so how do you find the sin of 135 degrees
00:19:14.860 --> 00:19:17.820
well first you can say is where is 135 degrees
00:19:18.940 --> 00:19:19.820
well this is 0
00:19:21.660 --> 00:19:25.340
this is 90, 135 is somewhere in here
00:19:27.620 --> 00:19:29.700
okay we dont care exactly where
00:19:29.960 --> 00:19:31.480
in terms of the picture
00:19:32.580 --> 00:19:34.660
you just know between 90 and 180
00:19:34.660 --> 00:19:39.740
okay now you say i need to construct a right triangle, and you always make a right triangle the x axis
00:19:43.260 --> 00:19:44.860
that angle is 45 degrees
00:19:45.560 --> 00:19:48.440
how do i know its 45 degrees? well this is 180
00:19:49.800 --> 00:19:50.300
that
00:19:51.700 --> 00:19:53.220
180, 180 minus 45 is 135
00:19:56.180 --> 00:19:59.140
the sin of 135 would be the same as the sin of 45
00:20:03.960 --> 00:20:04.840
the sin of 145
00:20:07.160 --> 00:20:08.440
is the same as sin 45
00:20:08.440 --> 00:20:13.440
then you have to figure out if its gonna be positive or negative, well its in the second quadrant
00:20:14.480 --> 00:20:15.280
you got our x
00:20:16.840 --> 00:20:17.340
for sin
00:20:17.700 --> 00:20:19.780
so its positive, for seawolves
00:20:19.840 --> 00:20:21.680
we had some others didnt we?
00:20:25.400 --> 00:20:27.000
well all these students
00:20:31.700 --> 00:20:35.540
so this is positive and we know that this is radical 2 over 3
00:20:37.940 --> 00:20:39.540
make sure you can do that
00:20:42.020 --> 00:20:42.660
tan of 330
00:20:43.920 --> 00:20:45.040
well 330 degrees
00:20:46.820 --> 00:20:47.620
is over here
00:20:49.100 --> 00:20:50.220
almost after 360
00:20:53.600 --> 00:20:57.920
you have the triangle, you say whats the angle between 330 and 360
00:20:58.400 --> 00:20:59.120
30 degrees
00:21:01.720 --> 00:21:03.640
so the tangent of 330 degrees
00:21:06.140 --> 00:21:07.180
will be the same
00:21:08.100 --> 00:21:09.860
as the taangetn 30 degrees
00:21:09.860 --> 00:21:12.020
now you have to check if its positive or negative
00:21:12.600 --> 00:21:14.200
so you look up the in the t
00:21:16.160 --> 00:21:17.200
and t stands for
00:21:19.100 --> 00:21:19.600
tribe
00:21:26.020 --> 00:21:28.100
oh c sorry i apologize, were in c
00:21:28.340 --> 00:21:29.460
which stands for
00:21:32.360 --> 00:21:32.860
cosine
00:21:35.580 --> 00:21:37.100
tangent is not a cosine
00:21:37.100 --> 00:21:39.140
so tangents suppose to be negative
00:21:39.740 --> 00:21:41.740
it will be negative tangent 30
00:21:42.000 --> 00:21:43.040
thats negative
00:21:44.300 --> 00:21:45.100
radical3/3
00:21:45.980 --> 00:21:47.580
or negative 1/radical 3
00:21:47.640 --> 00:21:50.920
you cant see that but thats okay thats what i wrote
00:21:55.100 --> 00:21:57.420
thats a negative square root of 3/3
00:21:59.660 --> 00:22:00.620
so far so good?
00:22:01.920 --> 00:22:02.420
2 for 2
00:22:06.220 --> 00:22:06.720
lets do
00:22:09.200 --> 00:22:10.000
225 degrees
00:22:15.360 --> 00:22:17.840
where is 225 degrees, well this is 180
00:22:20.580 --> 00:22:21.220
thats 270
00:22:23.320 --> 00:22:24.520
so its 35 down here
00:22:26.320 --> 00:22:28.000
but the angle between 225
00:22:29.480 --> 00:22:30.840
and 18- is 45 degrees
00:22:32.280 --> 00:22:33.560
so the cosine of 225
00:22:38.360 --> 00:22:40.280
is the same as the cosine of 45
00:22:41.440 --> 00:22:45.280
but were in the third quadrant and the tangent is negative
00:22:45.900 --> 00:22:47.420
the cosine is negatice
00:22:48.620 --> 00:22:49.900
so this is negative
00:22:51.960 --> 00:22:53.160
square root of 2/2
00:22:54.120 --> 00:22:54.620
3 for 3?
00:22:56.400 --> 00:22:56.900
3 for 3
00:23:01.300 --> 00:23:02.820
i want to see only yes's
00:23:16.480 --> 00:23:17.840
number 4 210 degrees
00:23:19.640 --> 00:23:20.600
well thats 180
00:23:21.460 --> 00:23:23.380
10 is somewhere around there
00:23:26.060 --> 00:23:26.780
so weve got
00:23:28.480 --> 00:23:29.440
33 is left over
00:23:33.240 --> 00:23:34.360
so the sin of 210\
00:23:35.480 --> 00:23:36.920
is the same as sin of 30
00:23:40.460 --> 00:23:42.220
over in the third quadrant
00:23:42.640 --> 00:23:43.680
sin is negative
00:23:47.180 --> 00:23:48.540
it goes negative 1/2
00:23:51.760 --> 00:23:53.280
you get that one right?
00:23:53.280 --> 00:23:54.700
you get the negative sign?
00:23:55.660 --> 00:23:56.700
you got the 1/2?
00:23:58.040 --> 00:23:58.540
good
00:23:59.900 --> 00:24:00.400
okay
00:24:02.340 --> 00:24:03.460
and cosine of 240
00:24:15.900 --> 00:24:18.460
240 means there is 60 degrees left over
00:24:21.600 --> 00:24:22.640
so cosine of 240
00:24:25.780 --> 00:24:27.300
is the same as cosine 60
00:24:27.760 --> 00:24:31.120
were in the tird quadrant so the cosine is negative
00:24:31.900 --> 00:24:33.500
the cosine of that is 1/2
00:24:35.020 --> 00:24:35.900
its also -1/2
00:24:36.720 --> 00:24:39.360
one thing you want to make sure you can do
00:24:39.360 --> 00:24:42.720
is the sin cosine and tangent of all of those angles
00:24:45.640 --> 00:24:47.240
pther trig stuff we have
00:24:50.580 --> 00:24:53.220
what about the traingle 0,180, and 270?
00:25:34.500 --> 00:25:37.060
lets draw the circle again for a second
00:25:54.480 --> 00:25:55.520
this is 0 degrees
00:25:56.020 --> 00:25:58.500
what are the oordinates at 0 degrees?
00:25:59.900 --> 00:26:02.220
the coordinates at 0 degrees is 1,0
00:26:03.180 --> 00:26:04.060
that tells us
00:26:05.000 --> 00:26:06.280
the cosine of 0 and 1
00:26:08.380 --> 00:26:09.340
and the sin of 0
00:26:10.260 --> 00:26:10.760
is 0
00:26:12.680 --> 00:26:15.720
what are the coordinates up here at 90 degrees
00:26:17.920 --> 00:26:18.420
0,1
00:26:18.780 --> 00:26:21.820
so it tells us that cosine of 90 degrees is zero
00:26:22.520 --> 00:26:24.440
and the sine of 90 degrees is 1
00:26:28.820 --> 00:26:29.780
so far so good?
00:26:33.000 --> 00:26:34.120
and how bout here
00:26:35.680 --> 00:26:36.480
180 degrees
00:26:40.880 --> 00:26:41.380
-1,0
00:26:44.060 --> 00:26:45.740
the cosine of 180 degrees
00:26:48.700 --> 00:26:50.620
is -1 and sin of 90 degrees is 0
00:26:53.840 --> 00:26:55.280
and now were down here
00:26:59.040 --> 00:26:59.840
270 degrees
00:27:01.320 --> 00:27:02.680
and thats point 0,-1
00:27:05.640 --> 00:27:07.560
the cosine of 270 degrees is 0
00:27:08.740 --> 00:27:10.420
and the sin of 270 degrees
00:27:13.580 --> 00:27:14.080
is -1
00:27:16.260 --> 00:27:16.760
so
00:27:18.280 --> 00:27:19.960
we can makeanother chart
00:27:44.540 --> 00:27:45.040
sin
00:27:46.380 --> 00:27:46.880
cosine
00:27:49.060 --> 00:27:49.620
tnagent
00:27:52.660 --> 00:27:53.380
sin of 0 is 0
00:27:56.580 --> 00:27:57.380
sin of 90 is 1
00:27:57.740 --> 00:27:58.940
cosine is reverse
00:28:01.100 --> 00:28:02.460
tangent sin/cosine
00:28:02.460 --> 00:28:04.360
and then you look and say wait a second
00:28:05.120 --> 00:28:07.120
tangent 90 degrees you get 1/0
00:28:08.000 --> 00:28:09.280
you cant divide by 0
00:28:11.600 --> 00:28:12.720
thats undefined
00:28:15.660 --> 00:28:17.820
then you do 180 and 270 and repeat
00:28:19.200 --> 00:28:20.240
o and -1, -1 and 0
00:28:22.860 --> 00:28:23.900
tangent is 0/-1
00:28:25.920 --> 00:28:26.480
and -1/0
00:28:27.060 --> 00:28:27.780
undefined
00:28:28.700 --> 00:28:32.140
its one of those places they trick you up in calculus
00:28:37.440 --> 00:28:40.320
working backward and going from 90 degrees
00:28:41.620 --> 00:28:44.260
we would worry about that next semester
00:28:45.980 --> 00:28:47.900
okay thats all the unit stuff
00:28:49.500 --> 00:28:51.180
now remember the radains
00:28:53.040 --> 00:28:55.040
can i show you what radians are
00:29:16.240 --> 00:29:18.560
so remember once around the circle
00:29:21.440 --> 00:29:22.400
is 360 degrees
00:29:22.400 --> 00:29:25.900
you say whats the distance when you go around the circle
00:29:27.560 --> 00:29:30.360
well if the radius is 1, the distance is 2pi
00:29:31.960 --> 00:29:33.560
we say thats 2pi radians
00:29:36.920 --> 00:29:39.960
or if you take degrees and turn it into radains
00:29:43.300 --> 00:29:44.260
would be 2pi/2
00:29:44.260 --> 00:29:45.920
you would say 180 degrees
00:29:47.020 --> 00:29:47.740
pi radians
00:29:53.040 --> 00:29:56.640
so that enables us to convert, so if you want to convert from
00:29:58.320 --> 00:29:58.960
degrees]
00:30:03.080 --> 00:30:03.800
to radians
00:30:07.280 --> 00:30:08.480
you take the angle
00:30:10.020 --> 00:30:10.520
theta
00:30:10.840 --> 00:30:12.040
youmultiply it by
00:30:13.360 --> 00:30:14.640
pi over 180 degrees
00:30:18.200 --> 00:30:21.480
and if you want to convert from radians to degrees
00:30:33.620 --> 00:30:34.980
you take angle theta
00:30:35.720 --> 00:30:37.400
you multiply it by the 180
00:30:38.700 --> 00:30:39.200
over pi
00:30:41.520 --> 00:30:42.240
so much fun
00:30:44.580 --> 00:30:46.020
how we doing on these?
00:30:46.780 --> 00:30:47.500
all of you?
00:30:56.220 --> 00:30:57.340
so lets practice
00:30:58.640 --> 00:30:59.920
i can cover this up?
00:31:02.600 --> 00:31:04.600
lets convert those to radians
00:31:09.900 --> 00:31:11.420
did you convert these?
00:31:13.800 --> 00:31:15.400
they are easy to convert
00:31:15.560 --> 00:31:16.360
first of all
00:31:17.240 --> 00:31:18.440
you can literally
00:31:19.260 --> 00:31:19.760
guess
00:31:27.160 --> 00:31:29.480
you literally can write 135pi/180
00:31:29.480 --> 00:31:30.840
but thats not all you have to do'
00:31:34.640 --> 00:31:36.400
how would i simplify this?
00:31:38.360 --> 00:31:38.860
3pi/4
00:31:42.560 --> 00:31:43.440
all pi over 4s
00:31:44.180 --> 00:31:45.540
are 45 degree angles
00:31:48.400 --> 00:31:48.900
330
00:31:52.220 --> 00:31:53.100
is 330pi/180
00:31:54.960 --> 00:31:57.520
you do a little dividing you get 11pi/6
00:32:02.700 --> 00:32:03.500
225 degrees
00:32:07.520 --> 00:32:08.400
is 225pi/180
00:32:09.920 --> 00:32:10.880
which is 5pi/4
00:32:14.380 --> 00:32:14.880
210
00:32:16.900 --> 00:32:17.780
is 210pi/180
00:32:21.540 --> 00:32:22.040
7pi/6
00:32:27.020 --> 00:32:27.520
and 240
00:32:36.300 --> 00:32:36.800
is 4pi
00:32:37.100 --> 00:32:39.900
over 3, were we able to convert all of those
00:32:40.880 --> 00:32:42.640
this is all making me happy
00:32:55.100 --> 00:32:55.900
almost done
00:33:03.000 --> 00:33:05.080
alright now im gonna try a graph
00:33:05.700 --> 00:33:06.660
sin and cosine
00:33:10.640 --> 00:33:14.080
with graph sin and cosine you start plotting points
00:33:14.540 --> 00:33:16.540
and you start plotting points
00:33:17.840 --> 00:33:20.960
youd get a graph that looks something like that
00:33:22.860 --> 00:33:24.860
thats a sin curve or a sine wave
00:33:52.280 --> 00:33:54.760
thats what the graph y equals sine x looks liek
00:33:59.040 --> 00:33:59.920
you said to me
00:34:00.740 --> 00:34:03.940
give me a certain number of degrees with the sine
00:34:03.940 --> 00:34:07.220
you can find it by finding the coordinates on the graph
00:34:07.820 --> 00:34:10.300
this graph can go on forever by the way
00:34:10.300 --> 00:34:14.220
because you can keep going around in a circle for as long as you want
00:34:14.940 --> 00:34:17.260
this is what a sine graph looks like
00:34:19.540 --> 00:34:20.100
sin pi/2
00:34:20.400 --> 00:34:22.480
thats where the sin pi value is 1
00:34:23.540 --> 00:34:24.420
sine of 3pi/2
00:34:24.460 --> 00:34:26.060
the lowest you have is -1
00:34:29.040 --> 00:34:29.920
0 at 0, pi at pi
00:34:30.480 --> 00:34:32.160
goes up and down like that
00:34:33.140 --> 00:34:33.860
very simpe
00:34:34.360 --> 00:34:35.720
nice pretty picture
00:34:39.040 --> 00:34:42.320
you can do a cosine graph, give you something to do
00:34:43.720 --> 00:34:44.840
you can do review
00:34:50.380 --> 00:34:52.300
okay lets draw a cosine graph
00:35:32.180 --> 00:35:34.740
thats what the cosine graph looks like
00:35:35.060 --> 00:35:36.180
the cosine graph
00:35:36.760 --> 00:35:37.880
is the same thing
00:35:38.080 --> 00:35:39.200
as the sine graph
00:35:39.200 --> 00:35:42.780
just started at a different spot and ended at a different spot
00:35:45.380 --> 00:35:47.540
so they are really the same graph
00:35:48.840 --> 00:35:51.880
but they are just shifted, shifted 90 degrees
00:35:52.620 --> 00:35:57.660
we did some transforming with this stuff before but we wont worry about that
00:36:01.220 --> 00:36:03.380
that can complete trigonometry
00:36:09.940 --> 00:36:12.100
what did you think i was gonna do?
00:36:14.560 --> 00:36:16.720
alright time to learn something
00:36:17.320 --> 00:36:19.880
so all this stuff we do in trigonometry
00:36:19.880 --> 00:36:21.600
is done with right triangles
00:36:22.680 --> 00:36:26.360
so what would happen if you didnt have a right triangle?
00:36:35.320 --> 00:36:37.800
suppose you just had any old triangle
00:37:04.500 --> 00:37:07.940
lets label the angles, capital a capital b capital c
00:37:08.680 --> 00:37:11.080
and the sides opposite those angles
00:37:11.600 --> 00:37:13.200
with lower case letters
00:37:14.140 --> 00:37:16.540
lower case a lower case b lower case c
00:37:30.860 --> 00:37:32.220
so what is the sin of a
00:37:33.440 --> 00:37:35.440
well if i drop a perpendicular
00:37:35.500 --> 00:37:37.020
i have right triangles
00:37:38.120 --> 00:37:39.020
and you know that the sin of a
00:37:43.240 --> 00:37:43.740
is h/c
00:37:48.480 --> 00:37:50.080
and you know that sin of c
00:37:53.960 --> 00:37:54.460
is h/a
00:37:58.600 --> 00:37:59.560
so far so good?
00:38:03.940 --> 00:38:05.460
now if i cross multiply
00:38:06.480 --> 00:38:08.000
your gonna get c sin of a
00:38:10.840 --> 00:38:12.360
a and here i get asin of c
00:38:13.820 --> 00:38:14.700
also equals h
00:38:14.700 --> 00:38:16.800
okay the factor that they equal eachother
00:38:20.220 --> 00:38:20.780
and i get
00:38:22.900 --> 00:38:23.400
sin of a
00:38:24.880 --> 00:38:25.440
equals a
00:38:26.840 --> 00:38:27.340
sin of c
00:38:31.540 --> 00:38:33.540
you have divide both sides by a
00:38:33.540 --> 00:38:35.680
by little a, divide both sides by little c
00:38:37.260 --> 00:38:38.860
i would get the sin of a/a
00:38:40.000 --> 00:38:40.880
is the sin of c
00:38:41.780 --> 00:38:42.280
over c
00:38:42.280 --> 00:38:44.920
cross multiply in your head and see if its true
00:38:49.180 --> 00:38:52.700
and i have plenty more to show that to equal the sin of c
00:38:55.700 --> 00:38:56.200
over b
00:38:56.860 --> 00:38:58.780
this is called the law of sine
00:38:58.780 --> 00:39:00.580
and you know you would have law
00:39:00.580 --> 00:39:01.080
of sin
00:39:02.820 --> 00:39:06.740
i will rewrite this now, can i show you where this came from?
00:39:07.260 --> 00:39:11.740
you shouldnt just take my word for it, firgure out where it came from
00:39:24.080 --> 00:39:27.040
this will only work with non right triangles
00:39:28.880 --> 00:39:30.000
we teach you guys
00:39:30.000 --> 00:39:33.760
it doesnt work for right triangles you dont need them for right triangeles, thats what sohcahtoa is for
00:39:34.220 --> 00:39:37.420
you need this when you dont have a right triangle
00:39:53.520 --> 00:39:54.720
this triangle abc
00:39:56.320 --> 00:39:58.320
lower case letters on the side
00:39:58.500 --> 00:40:00.420
these letters are the angles
00:40:01.760 --> 00:40:03.440
take the sin of each angle
00:40:04.840 --> 00:40:06.760
divided by the top of the side
00:40:07.840 --> 00:40:10.800
the top of the other angle divided by the side
00:40:13.000 --> 00:40:14.200
the opposite side
00:40:18.180 --> 00:40:19.300
it is very useful
00:40:23.460 --> 00:40:27.860
this is very useful because thats it you dont need right triangkes
00:41:02.460 --> 00:41:03.900
what if i want to find x
00:41:10.260 --> 00:41:11.220
thats 45 and 30
00:41:12.080 --> 00:41:14.640
thats 105, this is not a right triangle
00:41:18.560 --> 00:41:21.840
well now i remember my rule, i know that the sin of 45
00:41:24.780 --> 00:41:25.580
divided by x
00:41:28.240 --> 00:41:28.800
is sin 30
00:41:30.600 --> 00:41:31.480
divided by 12
00:41:36.200 --> 00:41:39.000
okay, a couple different ways to solve tha
00:41:40.340 --> 00:41:42.580
the first thing is cross multiply
00:41:51.920 --> 00:41:53.040
12sin45=xsin30
00:41:53.880 --> 00:41:55.160
and divide by sin 30
00:41:57.180 --> 00:41:59.020
you get 12 times the sin of 45
00:42:00.440 --> 00:42:01.320
over sin of 30
00:42:03.300 --> 00:42:03.800
x
00:42:08.040 --> 00:42:09.960
what is the sin of 45, well 2/2
00:42:14.380 --> 00:42:15.580
and sin of 30 is 1/2
00:42:18.600 --> 00:42:20.600
so thats 6 radical 2 over a half
00:42:23.780 --> 00:42:24.580
12 radical 2
00:42:26.560 --> 00:42:28.800
there you go you have to figure out
00:42:29.320 --> 00:42:30.920
what the missing side is
00:42:36.980 --> 00:42:40.660
sin and cosine cannot be used find cosines on wednesday
00:42:40.940 --> 00:42:42.780
but you can use these to find
00:42:42.780 --> 00:42:45.680
the sides of non right traingles or the angles
00:42:45.680 --> 00:42:47.060
of non right triangles
00:42:55.000 --> 00:42:59.000
cant do much with this without a calculator besides set it up
00:43:07.200 --> 00:43:07.700
ready
00:43:50.860 --> 00:43:52.540
suppose i wanted to find x
00:43:55.360 --> 00:43:56.400
us the law of sin
00:43:57.660 --> 00:43:58.860
so we know that the
00:44:01.900 --> 00:44:03.260
the sin of 60 is over x
00:44:05.900 --> 00:44:07.180
the sin of 45 over 16
00:44:07.180 --> 00:44:08.920
by the way you can write these the other way
00:44:09.180 --> 00:44:10.540
ypu could also write
00:44:12.980 --> 00:44:13.780
x over sin60
00:44:14.740 --> 00:44:17.060
is 16 over sin45, you can do that too
00:44:17.420 --> 00:44:20.220
doesnt matter as long as youre consistent
00:44:23.380 --> 00:44:26.020
anyway cross multiply, you get 16sin60
00:44:28.980 --> 00:44:29.620
is xsin45
00:44:31.700 --> 00:44:32.420
you divide
00:44:33.560 --> 00:44:34.600
you get 16sin60
00:44:37.920 --> 00:44:38.640
over sin 45
00:44:40.520 --> 00:44:41.080
equals x
00:44:42.560 --> 00:44:44.480
okay and then you just plug in
00:44:47.060 --> 00:44:48.500
sin of 60 is radicl 3/2
00:44:50.900 --> 00:44:52.180
sin45 is radical /2
00:44:55.380 --> 00:44:57.940
you get 8radical 3over radical 3 over 2
00:45:00.460 --> 00:45:02.300
16 radical 3, over radical 2
00:45:02.560 --> 00:45:04.480
you could of left it like this
00:45:04.880 --> 00:45:07.120
you know any of those forms is fine
00:45:07.240 --> 00:45:10.040
you dont need to rationalize, the truth is
00:45:12.700 --> 00:45:16.140
we probably wont give you happy angles like 60 and 45
00:45:16.140 --> 00:45:18.220
but youd probably have to leave it in that form
00:45:20.180 --> 00:45:21.780
can i rationalize that?
00:45:23.060 --> 00:45:24.260
16 radical 6 over 2
00:45:29.920 --> 00:45:30.420
8 radical 6
00:45:31.660 --> 00:45:32.380
8 radical 6
00:45:34.060 --> 00:45:36.380
howd you do on this one get it right?
00:45:38.260 --> 00:45:41.140
get it right? did you sorta just wait for me?
00:45:42.580 --> 00:45:46.100
how do we feel about law of sin? not really hard right?
00:45:49.320 --> 00:45:55.000
you can leave it like that or you can rationalize it some more, you can take 16 radical 3
00:45:58.720 --> 00:45:59.680
over radical 2
00:45:59.680 --> 00:46:02.380
and you multiple top and bottom by radical 2
00:46:03.760 --> 00:46:05.280
and you get 16 radical 6
00:46:05.980 --> 00:46:07.740
radical 2 and radical 2 is 2
00:46:08.960 --> 00:46:10.320
and thats 8 radical 6
00:46:10.680 --> 00:46:12.440
so all of those are the same
00:46:17.520 --> 00:46:19.120
howd we feel about this?