Stony Brook MAT 123 Fall 2015
Lecture 20: Trigonometry review, Law of Sines
November 2, 2015

Start   so first most of you trigonometry, make sure everyone remembers their trigonometry the first thing is sin, cosine ad tangents so what is sine of an angle the sin of angles and soh cah toa you have a right triangle you have the angle x
0:30you have 3 sides, you have side opposite you have the side adjacent and you have the hypothenuse and the sine of the angle is the ratio of the opposite sides and hypothenuse, remember al the trigonometry stuff should have taught you this the ratio of he triangle stays the same even if the size of the triangle changes
1:02if you have a right triangle and a particular angle, the ratio never changes we give names for the ratios to make it easier o work with they are called the sinx the ratio of this opposite side and the hypothenuse and then cosine of x is the adjacent side over the hypothenuse, and the tangent
1:33is the opposite over the adjacent and you have to remember all that sohcahtoa thats up on the instagram you guys all know this stuff right?
so if you didnt know it the last time around, this is a good chance to do it cause were gonna test you on this stuff again
2:11in the certain magic triangles that you work with one of the kind we work with is the 45, 45 90 and the 45, 45, 90 triangle
2:30comes from cutting the square in half on the diagonal if you cut a square in half on the diagonal, these two sides are still equal, still a square and that side is x radical 2 the sine of that angle, sorry that angle is always 45 degrees and that angle is 45 degrees you want to find the sign of 45 degrees
3:00its just the ratio of the opposite side and the hypothenuse now if you cancle the x's is 1/square root of 2 and then if you rationalize that multiply top and bottome by 2 it would also equal 2/2 this is stuff you want to make surely in your brain
3:31because when you do a lot of this in the next month, plus those lucky enough to take calculus you see a lot of trignometry in calculus i had a problem today, to find which sine and cosine intersect sine and cosine are the same sine and cosine are the same, theyre 45 degree angkes youre suppose to remember that all the way till MAT126 and of course
4:00so when youre 50 years old, really old and youre walking down the street and someone said whats the cosine of 45 degrees dont get a ride before you do i who will still be a live at that point, would say thank you very much maybe well i hope so im still oyung exactly cosine of 45 degrees is also x/x radical 2 why are they the same?
because these 2 sides are the same
4:32and thats also the square root of 2 radical 2/2 the tangent of x, x is 45 well the two sides are the same, the opposite and the adjacent he tangent of 45 is 1 thats how youre gonna master the chart you want to make sure you know these
5:02so far so good? cause im gonna cover this up in a minute now were gonna do 30, 60, 90 triagnle
5:36now suppose you take and equilateral triangle and you cut it in half, well and equilateral triangle when you cut it in half you divide the base in half. the base and this side is the same as a equilateral triangle this is x and this is 2x you can write the hypothenuse and this comes out
6:03x square root of 3 thats why we know alll the time so if i want to find the sine of 30 degrees you get x x to the 2x is 1/2 and the cosine of 30 degrees
6:35is x radical 3 over 2x which is radical 3/2 and the tangent of 30 degrees is x over x times the square root of 3 1/square root of 3 or if you want to rationalize radical 3 over 3
7:01make sure that youre comfortable with radical 3 over 3 and 1/ square root of 3 you know both are the same thing just like radical 3/2 and 1 over square root of 2 radical x over x is the same as 1/ square root of x radical 9 well radical 10 over 10 is the same as 1/ radical 10 they are all the same
7:35now if i want to do 60 degrees well the sine of 60 degrees, thats a 60 degree angle is the opposite over the hypothenuse, thats the smae as the cosine of 30 degrees and the cosine of 60 degrees
8:00is the same as sine 30 degrees the tangent of 60 degrees is x radical 3/x just comes out radical 3 you got all this down?
good cause im gonna make a little chart make sure you remember ill put the chart right heree so you can kind of see it
8:52so when i showed you before you can make this into a nice little chart
9:09where you can make this sin, cosine and tangent 30 45 and 60 degrees on top and then you can just fill in notice the sin of 30 degrees is 1/2 sin of 45 degrees
9:31radical 2/3 over here bit wait it was up there and now its down here alright so remember when you do the sine, everything is over 2 1, 2,3 and then for cosine everything is over 2 again
10:03you can do 3 2 and 1 thats a very good way to memorize the sine and cosines the tangent is found by taking the sine and dividing it by the cosine so sine over cosine is 1 cosine/ sine is square root of 3 alright that was our basic right triangle trick now theres other stuff you should know about right triangle ratio
11:00see that right there?
so were gonna do the pythagorean theorem so then for this triangle, a square plus b squared is c squared now what if i take everything and divide it by c squared well now i got a squared over c square or a over c
11:30squared b/c squared is 1 what is a/c, a/c is the sine of x b/c is the cosine of x the sine squared plus the cosine squared always equals 1 as long as they are the same angle
12:03so thats another thing you want to make sure you remember were gonna do trig identities in a week, so you got to make sure you remember these another thing is notice the tangent of x well thats a/b
12:32but if you took a and divide it by c and b/c you still have a.b and a/c is the sin and b/c is the cosine so the tagent of any angle is the sine of the angle divided by the cosine of the angle
13:02remember all this so far?
hope so for those of you who didnt do well on minimum competent on the first midterm this is a good time to remember this stuff thats another identity you want to make sure you know next thing we did we call this trig stuff and we used this to find angles
13:31other then your 90 degrees so how do you do that?
you draw a circle with the radius 1 one is also known as so this is our unit circle say if you want to find, sin and cosine what you do you can imagine radius is coming around the circle if you look at the angle and the radius
14:03the x axis this distance, oh lets call that theta this distance is x this distance is y that gives us the coordinates x,y but see this is 1 the sin of theta is y/1
14:32and the cosine of theta is x/1 so the coordinates is also just cosine theta, sin theta what you can do is imagine going aroun in a circle and where ever you stop the coordinates of that point are the cosine of theta around where you stop and the sine of the theta where you stop thats why you call it being a circle so you start here go around to some point and stop here
15:00and these coordinates are the sine that and the cosine of that thats why we use that that was this trick that that enable people to do sin cosine of angles other then between 0 and 90 degrees but they said i cant have a right triangle of an angle of 260 degrees you have to find another way to find the sin cosine of that so thats means we have to be able to shift quadrants
15:34and to find, sin and cosine so if you want to find the sin cosine of somethng you take the quadrant and you just figure out what the triangle would be and relate it back to the first one and you look and notice the coordinates change there signs, so in the first quadrant x and y are both positive so the sin and cosine are both positive in the second quadrant the x coordinate is negative the y coordinate stays positive
16:01so the sign will stay positive but the cosine will be negative in the third quadrant x and y are both negative down here so both sin and cosine will be negative and in the 4th quadranr dow here x is positive and y is negative so cosine will be positive and y is negative so alittle thing to help us remember that
16:33that reminds us whats positive and what negative so the first quadrant, so if you have an angle in the first quadrant so between 0 and 90 degrees call this 0 degrees and this 90 degrees all of the trig functions are positive if you go to the second quadrant between 90 and 180 degrees now the sine stays positive, the sign stays positive because the y coordinate is still positive
17:01the x coordinate is now negative because its going left so only sine is positive if you go down from 180 to 270 degrees now the tangent becomes positive but the sin and cosine become negative why is the tangent positive?
when x is negative y is negative the ratio you get positive and then you go to the last quadrant
17:30fourth quadrant the cosine is positive, why is the cosine positive?
because the x coordinates are positive and the y coordinates are negative x is cosine theta and y is sine theta so far so good?
okay everybody remember all this stuff?
good so you all should be able to find a problem
18:04were gonna practice right now before we move onto the good stuff
18:35were gonna do those 5 right now you all should be able to do these, notice i havent done radians yet got to convert it into radains so if you want to do these each of these is the exact same technique
19:01so you want to make sure you are really good at this because this is were people fell down in minimum competence with these kind of problems so how do you find the sin of 135 degrees well first you can say is where is 135 degrees well this is 0 this is 90, 135 is somewhere in here okay we dont care exactly where in terms of the picture
19:32you just know between 90 and 180 okay now you say i need to construct a right triangle, and you always make a right triangle the x axis that angle is 45 degrees how do i know its 45 degrees? well this is 180 that 180, 180 minus 45 is 135 the sin of 135 would be the same as the sin of 45
20:03the sin of 145 is the same as sin 45 then you have to figure out if its gonna be positive or negative, well its in the second quadrant you got our x for sin so its positive, for seawolves we had some others didnt we?
well all these students
20:31so this is positive and we know that this is radical 2 over 3 make sure you can do that tan of 330 well 330 degrees is over here almost after 360 you have the triangle, you say whats the angle between 330 and 360 30 degrees
21:01so the tangent of 330 degrees will be the same as the taangetn 30 degrees now you have to check if its positive or negative so you look up the in the t and t stands for tribe oh c sorry i apologize, were in c which stands for
21:32cosine tangent is not a cosine so tangents suppose to be negative it will be negative tangent 30 thats negative radical3/3 or negative 1/radical 3 you cant see that but thats okay thats what i wrote thats a negative square root of 3/3 so far so good?
22:012 for 2 lets do 225 degrees where is 225 degrees, well this is 180 thats 270 so its 35 down here but the angle between 225 and 18- is 45 degrees
22:32so the cosine of 225 is the same as the cosine of 45 but were in the third quadrant and the tangent is negative the cosine is negatice so this is negative square root of 2/2 3 for 3?
3 for 3
23:01i want to see only yes's number 4 210 degrees well thats 180 10 is somewhere around there so weve got 33 is left over
23:33so the sin of 210\ is the same as sin of 30 over in the third quadrant sin is negative it goes negative 1/2 you get that one right?
you get the negative sign?
you got the 1/2?
good okay
24:02and cosine of 240 240 means there is 60 degrees left over so cosine of 240 is the same as cosine 60 were in the tird quadrant so the cosine is negative
24:31the cosine of that is 1/2 its also -1/2 one thing you want to make sure you can do is the sin cosine and tangent of all of those angles pther trig stuff we have what about the traingle 0,180, and 270?
25:34lets draw the circle again for a second this is 0 degrees what are the oordinates at 0 degrees?
the coordinates at 0 degrees is 1,0
26:03that tells us the cosine of 0 and 1 and the sin of 0 is 0 what are the coordinates up here at 90 degrees 0,1 so it tells us that cosine of 90 degrees is zero and the sine of 90 degrees is 1 so far so good?
26:33and how bout here 180 degrees -1,0 the cosine of 180 degrees is -1 and sin of 90 degrees is 0 and now were down here 270 degrees
27:01and thats point 0,-1 the cosine of 270 degrees is 0 and the sin of 270 degrees is -1 so we can makeanother chart
27:44sin cosine tnagent sin of 0 is 0 sin of 90 is 1 cosine is reverse
28:01tangent sin/cosine and then you look and say wait a second tangent 90 degrees you get 1/0 you cant divide by 0 thats undefined then you do 180 and 270 and repeat o and -1, -1 and 0 tangent is 0/-1 and -1/0 undefined its one of those places they trick you up in calculus
28:37working backward and going from 90 degrees we would worry about that next semester okay thats all the unit stuff now remember the radains can i show you what radians are
29:16so remember once around the circle is 360 degrees you say whats the distance when you go around the circle well if the radius is 1, the distance is 2pi
29:31we say thats 2pi radians or if you take degrees and turn it into radains would be 2pi/2 you would say 180 degrees pi radians so that enables us to convert, so if you want to convert from degrees]
30:03to radians you take the angle theta youmultiply it by pi over 180 degrees and if you want to convert from radians to degrees
30:33you take angle theta you multiply it by the 180 over pi so much fun how we doing on these?
all of you?
so lets practice i can cover this up?
31:02lets convert those to radians did you convert these?
they are easy to convert first of all you can literally guess you literally can write 135pi/180 but thats not all you have to do'
31:34how would i simplify this?
3pi/4 all pi over 4s are 45 degree angles 330 is 330pi/180 you do a little dividing you get 11pi/6
32:02225 degrees is 225pi/180 which is 5pi/4 210 is 210pi/180 7pi/6 and 240
32:36is 4pi over 3, were we able to convert all of those this is all making me happy almost done
33:03alright now im gonna try a graph sin and cosine with graph sin and cosine you start plotting points and you start plotting points youd get a graph that looks something like that thats a sin curve or a sine wave
33:52thats what the graph y equals sine x looks liek you said to me
34:00give me a certain number of degrees with the sine you can find it by finding the coordinates on the graph this graph can go on forever by the way because you can keep going around in a circle for as long as you want this is what a sine graph looks like sin pi/2 thats where the sin pi value is 1 sine of 3pi/2 the lowest you have is -1 0 at 0, pi at pi
34:30goes up and down like that very simpe nice pretty picture you can do a cosine graph, give you something to do you can do review okay lets draw a cosine graph
35:32thats what the cosine graph looks like the cosine graph is the same thing as the sine graph just started at a different spot and ended at a different spot so they are really the same graph but they are just shifted, shifted 90 degrees we did some transforming with this stuff before but we wont worry about that
36:01that can complete trigonometry what did you think i was gonna do?
alright time to learn something so all this stuff we do in trigonometry is done with right triangles so what would happen if you didnt have a right triangle?
36:35suppose you just had any old triangle
37:04lets label the angles, capital a capital b capital c and the sides opposite those angles with lower case letters lower case a lower case b lower case c
37:30so what is the sin of a well if i drop a perpendicular i have right triangles and you know that the sin of a is h/c and you know that sin of c is h/a so far so good?
38:03now if i cross multiply your gonna get c sin of a a and here i get asin of c also equals h okay the factor that they equal eachother and i get sin of a equals a sin of c
38:31you have divide both sides by a by little a, divide both sides by little c i would get the sin of a/a is the sin of c over c cross multiply in your head and see if its true and i have plenty more to show that to equal the sin of c over b this is called the law of sine and you know you would have law
39:00of sin i will rewrite this now, can i show you where this came from?
you shouldnt just take my word for it, firgure out where it came from this will only work with non right triangles we teach you guys
39:30it doesnt work for right triangles you dont need them for right triangeles, thats what sohcahtoa is for you need this when you dont have a right triangle this triangle abc lower case letters on the side these letters are the angles
40:01take the sin of each angle divided by the top of the side the top of the other angle divided by the side the opposite side it is very useful this is very useful because thats it you dont need right triangkes
41:02what if i want to find x thats 45 and 30 thats 105, this is not a right triangle well now i remember my rule, i know that the sin of 45 divided by x is sin 30
41:30divided by 12 okay, a couple different ways to solve tha the first thing is cross multiply 12sin45=xsin30 and divide by sin 30 you get 12 times the sin of 45
42:00over sin of 30 x what is the sin of 45, well 2/2 and sin of 30 is 1/2 so thats 6 radical 2 over a half 12 radical 2 there you go you have to figure out what the missing side is
42:36sin and cosine cannot be used find cosines on wednesday but you can use these to find the sides of non right traingles or the angles of non right triangles cant do much with this without a calculator besides set it up
43:50suppose i wanted to find x us the law of sin so we know that the
44:01the sin of 60 is over x the sin of 45 over 16 by the way you can write these the other way ypu could also write x over sin60 is 16 over sin45, you can do that too doesnt matter as long as youre consistent anyway cross multiply, you get 16sin60 is xsin45
44:31you divide you get 16sin60 over sin 45 equals x okay and then you just plug in sin of 60 is radicl 3/2 sin45 is radical /2 you get 8radical 3over radical 3 over 2
45:0016 radical 3, over radical 2 you could of left it like this you know any of those forms is fine you dont need to rationalize, the truth is we probably wont give you happy angles like 60 and 45 but youd probably have to leave it in that form can i rationalize that?
16 radical 6 over 2 8 radical 6
45:318 radical 6 howd you do on this one get it right?
get it right? did you sorta just wait for me?
how do we feel about law of sin? not really hard right?
you can leave it like that or you can rationalize it some more, you can take 16 radical 3 over radical 2 and you multiple top and bottom by radical 2
46:03and you get 16 radical 6 radical 2 and radical 2 is 2 and thats 8 radical 6 so all of those are the same howd we feel about this?