Trigonometry Law of Sines / Sine Rule - Free Educational videos for Students in K-12

Trigonometry Law of Sines / Sine Rule - By tecmath

Transcript


00:00	Good day . Welcome to Tech Math channel . What
00:02	we're going to be having a look at in this
00:03	video is the sign rule . So sit back and
00:05	enjoy and before we start a big shout out to
00:07	my latest Patreon , Zoya . Your support is well
00:10	appreciated . Anybody who wants to become a Patreon and
00:13	support the Tech Math channel . You'll see a link
00:16	in the description below . So the sign rule .
00:18	This is a method used with non right angle triangles
00:21	. The triangle that we have here has no right
00:23	angles and it's used to work out unknown sides and
00:27	angles . Also , we can use the coastline rule
00:29	, which will have a look at in future videos
00:31	. So let's have a look at what the sign
00:34	rule is . A sign rule says the following .
00:36	It says that the sign of capital A . Over
00:39	little A . Is equal to the sign of capital
00:43	B . Over little B . Which is equal to
00:45	the sign of capital C . Over little C .
00:49	Or alternatively , the whole lot here can be flipped
00:52	as follows . We can have little A over the
00:55	sign of A . Is equal to little B .
00:59	Over the sign of B . Which is equal to
01:02	little See over the sign of C . It's a
01:05	nice easy rule to remember . Yeah . Um So
01:08	, what's this all mean ? Well , when we
01:10	label a triangle such as the triangle we have here
01:13	, we give the sides and angles specific letters .
01:16	And it's pretty easy . We label the angles here
01:19	with capital letters to the I'm going to call them
01:22	capital A capital B . And capital C . You're
01:26	going to see here . This angle A . Here
01:28	. This particular angle here opens out to this line
01:31	here , directly opposite little I . We call the
01:36	line the angle where angle B starts out there .
01:39	This line here , little B . And we call
01:41	this one , you guessed it ? This is going
01:43	to be seized . So for the law of science
01:45	, we're going to be using matching sets to work
01:47	out unknown sides . Okay . So I'll show what
01:50	I mean . And I'm just going to reduce this
01:52	rule here to its most simple by getting rid of
01:55	all this stuff here . So say I was just
01:58	considering sign A and A and sign B and B
02:01	. Angle a long and a angle B . And
02:04	line be . We have too much in sets here
02:07	. We could use to say we knew these particular
02:11	figures here . We knew the angle of am we
02:13	knew the line length of A here and say we
02:16	only knew be here . We could use this also
02:19	to work out this particular side length over here .
02:23	Okay . That's where the sign rule is really ,
02:25	really handy . As long as we know at least
02:27	one full set and part of the other set ,
02:29	we can use it to work out the rest .
02:31	Okay , so let's have a look at this in
02:33	action . I'm just going to go through two examples
02:35	. We're going to work out one with a side
02:37	, we're going to work out one with the angle
02:39	. And I'm also going to show you just a
02:40	little trick that you can use within this just in
02:43	case you get a particular type of question . Okay
02:46	, so Let's have a look at an example here
02:49	. Okay , for our first example here we have
02:51	a triangle where we have an angle of 37° and
02:55	its corresponding side that opens out to seven . We
02:58	also have an angle of 97° which opens out to
03:02	a corresponding side that we're trying to find . So
03:04	we're gonna use a condensed version of the sign rule
03:07	as follows . We're going to be using this particular
03:09	one here . A over the sign of capital a
03:13	equals B over the sign of capital B . We
03:17	only need to use this particular part of the soil
03:20	. We don't need to use the whole thing .
03:22	Okay , So let's go through and lay on their
03:26	sides . Now . Now I would like to keep
03:28	my unknowns on this side . It's just a little
03:30	thing I like to do . So let's start with
03:32	that . We're gonna unknowns here , which I'm going
03:34	to be calling . How easier . So this is
03:36	going to be a capital A for our angle and
03:39	a little A for our side which goes along with
03:41	it . We have a capital B for angle here
03:45	that we do know and a little B for that
03:47	side length of here that we do know . So
03:50	now let's go through and substitute in values . Okay
03:53	, So we have , okay , Which is equal
03:56	to X . We have Sign of 97°. . Okay
04:05	? And this is equal to be which is seven
04:08	Over a sign of 37°. . All right . Yeah
04:15	. We didn't go through now and work out the
04:16	sign of 97 and the sign of 37 . So
04:20	, if we were to do this , we're going
04:21	to get the following . We're gonna get the sign
04:23	of 97° And we're going to work out the sign
04:27	of 37°. . And these are as follows . A
04:30	sign of 97° is equal to 0.99 . The sign
04:36	of 37° is equal to 0.60 . So now we
04:41	can substitute these into our equation here . Okay .
04:44	So we're gonna end up with X over the sign
04:48	of 97°, , which is 0.99 , Which is equal
04:52	to seven Over the sign of 37°, , which is
04:56	0.60 . All right . We can solve this particular
05:00	party is seven divided by 0.60 So X over 0.99
05:07	is going to eagle . Whatever seven divided by 70.60
05:11	is which is 11.666666666 11.67 All right . What do
05:19	we do now to find X ? This is what
05:21	I usually do . I've done this in other videos
05:23	when we don't know what to do . We could
05:24	go a friendly equation . 6/2 equals three . We're
05:28	trying to find the one at the top here .
05:30	We're going to multiply these because two times three or
05:33	six we're going to multiply these two numbers to find
05:35	our next cycles . So , X is equal to
05:38	this number times this number which is equal to 11.55
05:44	. So this particular side idea is equal to 11.55
05:49	. And that's how easy it is to use the
05:51	sign role . For our next example , we're going
05:53	to be using a two step problem . We're going
05:55	to find an angle , but it's gonna be a
05:57	little bit of an extra step in there . So
05:59	let's get to that . Okay , for this particular
06:01	example here we have a non right angle triangle .
06:04	We have an angle of 44° that opens out to
06:08	a side length of 12 . We have an angle
06:10	here that we're gonna be trying to find here .
06:11	That opens out to a side length of 14 .
06:14	So we're going to be using the following formula .
06:16	We're gonna be using the sign of capital a .
06:19	I will little I equals the sign of capital B
06:23	over a little B . It's the angle . I
06:25	like to use the sign on top here . And
06:28	if it's a side link , I use the little
06:30	on top there . So just a little thing just
06:32	makes life a whole lot easier as you'll find out
06:34	as you go along . So let's work these out
06:37	. So let's label our sides first . I like
06:39	to keep the unknowns on this side . So I'm
06:40	going to be calling this side A . And this
06:43	side little A . I'm going to call this side
06:45	therefore capital B . And this side little B .
06:47	So let's sub in our values the Okay , so
06:50	a sign of A . Is equal to sine feta
06:55	. Okay . And little a . Here is equal
06:58	to 14 . Okay , this is equal to the
07:01	side of B . Which is Alright , we're sign
07:04	of v . Which has signed 44°. . And we're
07:08	gonna put this over Little Bit , which is 12
07:11	. So what is first off the sign of 44°?
07:15	? The sign of 44° is equal to you can
07:19	work this out on your calculator , 0.69 . So
07:23	let's sub this in to our equation here . We
07:27	have signed Feeder Over 14 which is equal to Uh
07:34	what do we say ? 0.69 over 12 . So
07:38	Writing it all out again now , signed Feeder over
07:41	14 is equal 2.69 , divided by 12 is 0.058
07:50	. Okay . So how we gonna watch sold for
07:53	sign feed here ? Well what before we could do
07:56	this old 6/3 equals two . Where after the number
08:00	on the top ? So we're going to multiply these
08:02	two numbers . We're going to multiply . These two
08:03	numbers were after the number on the top . Okay
08:06	? So sign feta is equal to 14 times 0.058
08:16	, Which is equal to 0.81 . So how we're
08:20	gonna work out what feta is now . We're just
08:23	going to work out a sign to the negative one
08:27	of 0.81 So we're going to find out the feta
08:31	is equal to the inverse of that , which is
08:34	54.14 degrees . So this angle here is 54.14 degrees
08:41	. And just before we go here , just a
08:43	little thing I want to tell you just to watch
08:45	out for I'm just going to leave . It is
08:46	54 for the minute but occasionally you'll be asked to
08:48	work out say this angle here and you'll be given
08:51	uh you know , you won't be asked to give
08:53	you this angle , I'll give you this side and
08:54	I'll ask you to work out instead this angle .
08:55	So you mustn't do that . I'm working at this
08:57	angle here , and then what I do is I
08:59	would say , well , OK , there's 100 and
09:01	80 degrees in a triangle . So this angle here
09:04	is going to equal to 180 take away 44 take
09:08	away 54 you'll have that particular angle here . So
09:13	, anyway , that's the sign rule . Hopefully you
09:15	liked it . Uh If you did , please remember
09:17	to like and subscribe and as I said before ,
09:20	if you really want to support the tech mouth channel
09:22	, become a Patreon will be really , really appreciated
09:24	. Anyway , next video we're going to have a
09:27	look at is the co sign rule . Uh Anyway
09:29	, I look forward to seeing you catch you next
09:31	time . Bye .

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