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Law of Sines

  • Law of Sines: Let ΔABC be any triangle with a, b, and c representing the measures of the sides opposite angles with measure A, B, and C, respectively. Then,

Law of Sines

Determine whether the following statement is true or false.
In a triangle, given the measures of two angles and the length of any side, we can find the lengths of the other two sides.

m∠A = 60o, m∠B = 45o, AC = 10, find BC.
  • [sinA/BC] = [sinB/AC]
  • [sin60/BC] = [sin45/10]
BC = [10sin60/sin45] = [10*0.866/0.643] = 13.5

m∠A = 30o, AC = 12, BC = 18, find m∠ B.
  • [sinA/BC] = [sinB/AC]
  • sinB = sinA*[AC/BC]
  • sinB = 0.5*[12/18]
  • sinB = 0.333
B = 19.5o
Determine whether the following statement is true or false.
In triangle ABC, a, b and c represent the measures of the sides opposite angles with measure A, B, and C, respectively, if sinA > sinB, then a > b.
Determine whether the following statement is true or false.
In a triangle, given the lengths of three sides and the measurement of one angle, we can find the measurements of the other two angles.

∆ABC, if AB = 20, BC = 16, m∠C = 60o, find m∠ B.
  • [sinA/BC] = [sinC/AB]
  • [sinA/16] = [sin60/20]
  • sinA = [16/20]*sin60
  • sinA = 0.693
  • m∠A = 44o
  • m∠B = 180 − m∠C − m∠A
  • m∠B = 180 − 60 − 44
m∠B = 76o.

m∠A = 90o, m∠B = 40o, m∠C = 50o, AC = 15, find AB and BC.
  • [sinA/BC] = [sinB/AC] = [sinC/AB]
  • AB = AC*[sinC/sinB]
  • AB = 15*[0.766/0.643] = 17.9
  • BC = AC*[sinA/sinB]
  • BC = 15*[1/0.643]
BC = 23.3.

determine whether the following statement is true or false.
If sinA = sinC, then AB ≅ BC .

solve ∆ABC.
AB = 24,BC = 18,AC = 15,m∠A = 60o.
  • [sinA/BC] = [sinB/AC] = [sinC/AB]
  • sinB = [AC/BC]*sinA
  • sinB = [15/18]*0.866 = 1.039
  • m∠B = 46.2o
  • m∠C = 180 − m∠A − m∠B
  • m∠C = 180 − 60 − 46.2
m∠C = 73.8o.

Trapezoid ABCD
m∠ACB = 45o, m∠B = 120o, AB = 12, m∠D = 60o, find CD.
  • [(sin∠ACB)/AB] = [sinB/AC]
  • AC = AB*[sinB/(sin∠ACB)]
  • AC = 12*[sin120/sin45] = 12*[0.866/0.707] = 14.7
  • m∠CAD = m∠ACB = 45
  • [(sin∠CAD)/CD] = [sinD/AC]
  • CD = AC*[(sin∠CAD)/sinD]
CD = 14.7*[sin45/sin60] = 12.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.


Law of Sines

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Law of Sines 0:20
    • Law of Sines
  • Law of Sines 3:34
    • Example: Find b
  • Solving the Triangle 9:19
    • Example: Using the Law of Sines to Solve Triangle
  • Extra Example 1: Law of Sines and Triangle 17:43
  • Extra Example 2: Law of Sines and Triangle 20:06
  • Extra Example 3: Law of Sines and Triangle 23:54
  • Extra Example 4: Law of Sines and Triangle 28:59

Transcription: Law of Sines

Welcome back to

For the next lesson, we are going to start going over some concepts to use for non-right triangles.0002

Everything that we went over so far in this unit had to do with right triangles, and now we are going to go over some non-right triangles.0010

The first one is Law of Sines: we already went over trigonometric functions (sine, cosine, and tangent); this has to do with sine, the first one.0018

The Law of Sines says that, when you have a non-right triangle ABC, with a, b, and c--0031

notice how they are lowercase--representing the measures of the sides opposite the angles with measures A, B, and C (capital A, B, C)0044

respectively, then this is true, right here; this is actually the Law of Sines.0054

Now, let's go over that again: triangle ABC's angles are capital letters--capital A, capital B, capital C.0064

Now, here the lowercase a, b, and c are representing the sides opposite those angles.0076

For angle A, the side opposite angle A, which is this side right here--we can name this as lowercase a.0084

And then, the side opposite angle C is going to be lowercase c, and the side opposite angle B will be lowercase b.0095

Whenever you see capital letters, they are going to be representing angles.0107

And lowercase letters are going to be representing the sides.0114

Since they are the same letters, just keep in mind that it has to be the angle with the side opposite; they have to go together.0119

In that case, now, remember again: this is for non-right triangles.0131

If we have a right triangle, then we would use the Pythagorean theorem; we would use special right triangles,0134

geometric mean, and Soh-cah-toa (trigonometric functions) to find unknown sides and angles--unknown measures.0143

This is used for non-right triangles; so if you have a triangle that is not a right triangle, then this can be an option to use.0155

Sine of A...remember, you can only find the sine of angle measures...over the side a, equals the sine of B (angle B), over side b;0167

that is equal to the sine of capital C, which is the angle C, over side c.0181

Now, you are not going to use all three; we are going to use two of the three to make a proportion, and then solve it out that way.0187

But it is just letting you know that all three ratios are equal to each other.0195

And then, you would just use two of the ratios that you need to find out the unknown value.0199

So again, sine of A over a equals sine of B over b, which equals sine of C over c.0205

Let's use the Law of Sines to find b, lowercase b; so we know that we are looking for the side opposite angle B; that is this right here.0216

Now, the Law of Sines says that the sine of A, over a, is equal to the sine of B, over b; and that is equal to the sine of C, over c.0229

So, here I have angle C and side c; I have this, and I have this.0245

And then, I have angle B, and I am looking for side b; so I would have to use this one, because here is my unknown value.0258

The sine of 85 (B is 85), over b, is equal to the sine of 50 degrees, over a.0271

Now, again, only angle measures go with the sine: sine is of angle measures only.0289

Then, this would be the side c, which was 8; there we have our proportion.0296

Now, we can solve for b, the one variable here.0301

Remember: these always have to go together; they can't split up.0309

Sine cannot go by itself; it is sine of an angle measure.0311

When we cross-multiply, I am going to write b times sin(50).0316

Do I multiply anything together? No, I have to keep sin(50) separate.0324

It, 8 times sin(85): again, do you multiply 8 by 85?0329

No, you have to keep that separate, because the sine of that angle measure is one number; and then whatever this is--then you can multiply that by 8.0338

But you cannot multiply 8 by 85; make sure that you don't do that, because,0349

if you multiply 8 by 85, then that no longer becomes the angle measure for sine.0354

You can do this two ways: you can solve these out on your calculator, and then solve for b;0362

or you can just solve for b first, and then use your calculator.0372

Here, take your calculator; you don't have to use a graphing calculator--you can use a scientific calculator--anything with those trigonometric functions.0378

The sine of 50--that is the first one: that is .7660, so this right here became .766, and then b is equal to 8 times that.0395

We are only solving out the trigonometric functions first: sin(85) get .9962.0418

Now, from here, you can multiply that by 8, and then divide this.0437

So, if I just go ahead and divide this, b is equal to all that; so then, you just use your calculator.0446

.9962...I don't have to erase it; I can just leave it on my calculator, and just multiply that by 8; it equals 7.96.0457

And then, from there, just go ahead and divide by .766, and you are going to get 10.4, so b is 10.4.0469

Again, you see what you have: you need two ratios to make a proportion.0489

We have angle C, and we have side c, so we are going to use that.0495

And then, b is what we are looking for, so we have to use this one.0501

Then, you just plug everything in; you are going to cross-multiply.0505

You can solve it out first if you would like; it doesn't matter.0509

You can calculate this out and then start cross-multiplying; or you can cross-multiply first, which is what I did, and then you can calculate this and this.0511

The reason why I calculated these out first was so that you wouldn't get confused when I started dividing this whole thing.0526

But you can divide this whole thing, because you want to solve for b (isolate b), so you can divide the whole thing by sin(50).0533

Just make sure that you divide this whole thing together; sin(50) has to stay together--that is one thing that has to stay together.0541

And then, it is all calculator from there.0550

Moving on: now, here we have what is called solving the triangle.0559

When you solve the triangle, you are looking for all unknown values.0568

A triangle, we know, has three sides and three angles.0576

So, minus what they give you--whatever value is unknown, you need to solve for.0580

Here there are going to be six in all, three angles and two sides.0590

But here, they give you angle A; they give you side b; and they give you side a.0597

Everything that you have to solve for would be the measure of angle C, the measure of angle B, and then...what else is missing?...side AB.0604

So, all three measures have to be solved for; they have to be found, and that is called solving the triangle.0622

To solve the triangle one thing at a time, what do I have? I have angle A, and I have side a.0635

I have side b, and I can find angle B using the Law of Sines.0645

Sine of A, over a, equals sine of B, over b, which is equal to sine of C, over c.0653

I am going to use this one, and I am going to use this one, because this is my unknown.0662

The sine of 45 (that is angle A), over 10, equals the sine I know B?, that is what I am looking for, so I leave it like that...over 14.0671

So then, I can cross-multiply this; and if you cross-multiply, you are going to get 10sin(B) = 14sin(45).0691

Now, again, don't multiply the 14 and the 45 together; do not multiply those numbers together.0710

Then, from here, because again, I am solving for B, so I can just divide this 10--get rid of that 10 and divide this whole thing by 10.0721

So, the sine of B, let's use our calculator from here.0730

The first thing I am going to do is solve for sin(45); sin(45) is .7071.0738

Then, multiply that by 14; you don't have to delete it off of your calculator--just multiply it by 14, and then divide by 10; and you get .9899.0749

Now, that is not our answer, because we are looking for this angle measure, B.0772

Now, remember that, when we punch this into our calculator...make sure that you don't do sine of this number,0777

because then the number you punch in, right after you push sin, is supposed to be the angle measure.0784

If you push sin and a number, that number must be the angle measure.0792

This is not the angle measure, so we can't punch it in right after sin.0796

Since we are looking for the angle measure (we don't have the angle measure; that is what we are looking for), we have to do inverse sine.0800

So then, push 2nd and sin; that way, on the top of your sin button, you see sin-1.0807

And then, you are going to punch in .9899, and by doing that, you will get the angle measure, which is 81.8 degrees.0818

The measure of angle B is 81.8 degrees; there is B--we found that one.0839

Now, to find the measure of angle C, we could do all this again.0849

Actually, I wouldn't be able to, because I don't have side c.0857

So, if I don't have angle C, and I don't have side c, then I can't do this.0861

But what you can do: the best way to do that is to subtract it from 180, because we know that all three angles of a triangle have to add up to 180.0866

So then, I am going to do 81.8 degrees, add it to 45, and then subtract that number from 180;0875

and then, you get that the measure of angle C is 53.15 degrees.0893

I have this, and I have this; and now I need to look for AB, which is side c.0903

Now, let me just write c, since that is how it is here: side c.0908

Now that I have this (this is 53.15 degrees), I can now go back to this and solve for my side by using this ratio.0916

Now, I have everything else, so I can choose this one or this one (it doesn't matter) for my second ratio.0928

I am going to just use A; so here it is going to be sin(45)/10 = sin...what is angle C?...51.15, over side c (is what I don't have; that is what I am looking for).0935

From here, cross-multiply: csin(45) = 10sin(53.15).0964

I can just go ahead and divide this whole thing by sin(45).0987

And then, just punch it in your calculator: first, sin(53.15) times 10, divided by sin(45)...and you get (this all canceled out) that side c is 11.317, or 11.3.0998

That is this measure right here; so again, from here, just punch in 10 times the sine of that,1042

divided by the sine of that, and you should get this answer right here.1048

Don't forget parentheses, if you need parentheses.1052

That is solving the triangle, just looking for all unknown measures.1058

OK, now let's do our examples: Find the measure of angle A here.1064

So again, we are going to just use the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c.1068

We don't have this; we have a; and then, we have the c's, angle C and side c.1084

I am going to use this one and this one: sine of angle A, over 16, equals sine of 62, over 21.1094

21sin(A) = 16sin(62); again, don't multiply these together.1113

Divide the 21; sin(A) = 16sin(62), divided by 21; that is .6727.1127

Now, I am looking for angle A; that means I have to do inverse sine, because it is not angle measure that I have.1161

So, I have to use inverse sine, 2nd and sin: .6727...the measure of angle A is 42.28 degrees.1171

And that is all we had to find, the measure of angle A.1202

Example 2: Here, they don't give you a triangle; they are just giving you these measures.1207

a is 10; the measure of angle B is 40; the measure of angle A is 55 degrees; find b.1215

We know that we are going to use sin(A)/a = sin(B)/b.1221

So, sine of...what is A?...the measure of angle A is 55 degrees, over 10,1232

equals the sine of...the measure of angle B is 40, over b (is what we are looking for).1240

To continue here: bsin(55) = 10sin(40); divide sin(55); don't separate these--sin and 55 have to go together.1249

Divide this by the sine of 55 degrees; you just take your calculator: 10 times the sine of 40, divided by the sine of 55: b = 7.8.1270

All of this b equals 7.8; just make sure that you multiply the sine of 40...punch in sin(40), multiply that by 10, and divide it by sin(55).1306

And the next one: here is side c, side b, angle B...and find angle C.1324

I am going to use sin(B) over side b equals sine of angle C, over side c.1333

The sine of B is 42, over 15, is equal to the sine of C (that is what we are looking for), over...side c is 12.1343

Cross-multiply: 15sin(C) = 12sin(42); divide the 15; sin(C) = 12sin(42)/15, so it is 0.5353.1361

I need to find my angle measure, so I am going to push 2nd and then sin, the inverse sine, of .5353.1404

The measure of angle C is 32.36 degrees.1418

We are going to solve the triangle; that means that we have to look for the measure of angle C; we have to look for side c and side a.1435

To find the measure of angle C, since we are given two angles--whenever you are given two angles, you just have to subtract them from 180.1452

52 plus 48 becomes 100; so 180 - 100 is 80 degrees--that is the measure of angle C.1463

And then, let's look for a and c; to find a, I'll do...let's just go straight into the Law of Sines.1479

The sine of angle A is 52, over side a, equals...we can't use this yet, so it would be sin(48)/9.5.1496

asin(48) = 9.5sin(52); divide the sine of 48; so a is going to be...find 9.5sin(52); and we divide this by sin(48); and that is 10.07.1518

a is going to be 10.07; now, let's look for side c.1571

We are going to do sine of AD, sine of C, over c, equals...well, let's use B; so sin(48) over 9.5.1579

c times the sine of 48 (cross-multiplying) equals 9.5 times sin(80).1596

Divide the sine of 48; c equals (it is all a calculator from here) 9.5 times sin(80), divided by sin(48); so I am going to get 12.6 here for c.1609

One way to check to see if your answers are correct: this is 80; this is 12.6; and this was 10.1 or 10.07;1651

angle B has the smallest measure--that means that side b should be the shortest side.1670

Next is angle A: the measure of angle A is 52, so it should be slightly bigger than side b, which is 10.07.1678

And then, angle C has the largest angle measure; this is the largest angle--that means that the side opposite it has to be the largest side.1690

So then, smallest to biggest: this side should be the smallest, and this side should be the biggest, depending on how big the angle measures are.1699

If we solve for c, and we get 20, or if we get 2, then we know that that is wrong, because,1709

since angle C has the largest angle measure, the side opposite has to be the largest side; it has to be the longest measure.1720

So, if it is smaller than any of these numbers, then you know that there is something wrong; you did something wrong.1729

It is just one way to check.1737

Your last one: Find the length of BC and BD.1740

Here is BC, and here is BD; so then, here I know that this is a parallelogram, because we have two pairs of opposite sides being parallel.1747

These two are parallel, and these two are parallel; and that is by definition of parallelogram that this is parallelogram ABCD.1764

Let's see, now: we know that we are on triangles, so we are going to be using something with these triangles here.1775

They want us to find BC (so then here, let's say this is x) and BD (let's say this is y).1783

Now, for this one here, let's see: here I know that this is going to be 8, because this is the side opposite,1795

and then, because these are congruent (in a parallelogram, opposite sides have to be congruent), what else do I know?1819

I know that this angle and this angle are congruent, because they are alternate interior angles (parallel alternate interior angles).1829

The same thing here: because these two lines are parallel, and this is a transversal, and this is 72, then this has to be 72.1845

And then, you can use those two: subtract from 180, and then that will be angle C; but I don't think we will need that.1858

Here, we can use the Law of Sines now to find this side: an angle with the opposite and an angle with this opposite.1867

The sine of 50, over 8, is going to equal the sine of angle measure 72, over x.1878

Cross-multiply: xsin(50) = 8sin(72); divide the sine of 50; so x is going to be 8 times the sine of 72, divided by sin(50).1894

So, I am going to get 9.9 for this length right here; and that is going to be BC--that was 9.9.1936

So then, that one is done; and then, to find BD, let's see; I do need that angle measure, because that is the angle opposite that side.1952

So here, this is going to be...50 + 72 is 122; 180 - 122 is going to be 58 degrees.1963

This is 58; to find the y, I can use any pair, so let's just use this one, because this is a whole number.1986

The sine of angle measure 50, over the side opposite, 8, equals the sine of 58, over y.1998

y times sin(50) equals 8 times sin(58); divide sin(50); y equals...and that becomes 8.86, or 8.9; and that will be BD; it is 8.9.2016

Now, why is it that, if we have this side here, and we have x (we have BC right there, 9.9)--2073

why can't we use the Pythagorean theorem to find this third side?2083

That is because this is not a right triangle; you can only use the Pythagorean theorem for right triangles.2087

This is the Law of Sines; you can only use it for non-right triangles, any time that you have a triangle where they all have to be acute angles.2096

They are acute angles; no right triangles; and you are looking for angle measures or side measures.2108

Then, you are going to be using the Law of Sines.2119

That is it for this lesson; thank you for watching