Q: I can't find anything here about ambiguous triangles. What if a question asks you to solve from a description where two triangles exist? Like "Determine the unknown side and angles in each triangle, if two solutions are possible, give both: In triangle ABC, <C = 31, a = 5.6, and c = 3.9." I solved for height and see that two solutions exist, and the answer key in my textbook agrees, but I can't figure out how to get either. From a set of questions that's only supposed to be on sine law.

Use the Law of Sines to get one possible angle A: sin(A)/a=sin(C)/c sin(A)/5.6=sin(31)/3.9 sin(A)=5.6sin(31)/3.9 A=arcsin(5.6sin(31)/3.9)=47.6924 Subtract 31 (C) and this angle (A) from 180 to find the third angle (B=101.3076) and use the Law of Sines again to find the third side. If you use the given angle-side pair (C and c) you will be less likely to incur error from your own rounding of angle A: b/sin(B)=c/sin(C) b/sin(101.3076)=3.9/sin(31) b=3.9sin(101.3076)/sin(31)=7.4253 But if you know that supplementary angles share a sine value, you know that A can also be an obtuse angle with the same sine as 47.6924: A=180-47.6924=132.3076 And again, subtract 31 (C) and the obtuse angle A from 180 to find the other possible third angle (B=16.6924) and use the Law of Sines to find the other possible third side, again using angle C and side c to avoid errors from rounding: b/sin(B)=c/sin(C) b/sin(16.6924)=3.9/sin(31) b=3.9sin(16.6924)/sin(31)=2.1750 It all comes from knowing that there are two angles, one obtuse and one acute, for every sine value. And you find the obtuse one by subtracting the acute one from 180. If you try to do this with a unique triangle (one without two possible sets of angles and sides) your given angle and the obtuse angle you find will add up to more than 180, and so if you try to find a third angle to go with the obtuse one, your subtraction will tell you the third angle is negative, at which point you know you're going down a nonsensical mathematical road, and there weren't two possible triangles to begin with. Good luck!

Q: In the Video 'Solving An Angle With The Law Of Sines' at 2:05, Sal said that Law of Sines is 'sin(a)/A = sin(b)/B = sin(c)/C (lower case letters are the angles and the upper case letters are the side opposite to the angle), in this article it says 'A/sin(a) = B/sin(b) = C/sin(c)'. So which one should I choose?

The Law of Sines can be written either way! You can put the angles in the numerators and the sides in the denominators, or the other way around. To understand why, think about this true equation: 1/2 = 2/4 = 3/6 It's still true if we reverse the numerators and denominators: 2/1 = 4/2 = 6/3

Q: Is this a correct way to check whether you use the law of sines or law of cosines? If you have two angles and a side or two sides and an angle, use the law of sines, if you have 3 sides use the law of cosines. Is there anything you need to add to this list?

The law of sines works only if you know an angle, a side opposite it, and some other piece of information. If you know two sides and the angle between them, the law of sines won't help you. In any other case, you need the law of cosines.

Q: If law of sines is a/sin(a)=b/sin(b)=c/sin(c), does sin(a)/a=sin(b)/b=sin(c)/c work?

Yes. Just raise one of those equations to the -1 power, and you get the other equation. They're equivalent.

Q: What is the numeric form of sine, cosine, and tangent? Like what does the calculator use to calculate trig functions?

The trig functions can be expressed as polynomials of infinite degree, called Taylor polynomials. For example, sin(x)=x-x³/6 +x⁵/120 -x⁷/5040+... Some calculators use these series to approximate the values of the trig functions, as well as other functions like exponentials. You'll develop Taylor polynomials in calculus.

Q: What is the law of tangents used for??

The law of tangents is used to keep people from getting to the point. Old people tend to use it a lot :)

Question 1

I want to know why this article says "Remember that if the missing angle is obtuse, we need to take 180 degrees and subtract what we got from the calculator" when using the law of sines to find a missing angle. Are there videos that explain why this is? I don't understand why during calculations it won't just give the obtuse angle, and instead gives an acute angle.

I'm only taking this geometry course and there is nothing along the way that explains this, not even the videos that introduce the law of sines.

Thanks.

I'm only taking this geometry course and there is nothing along the way that explains this, not even the videos that introduce the law of sines.

Thanks.

Accepted Answer

There can be two since sin(theta) = sin(180-theta) for all values of theta that are real numbers e.g. -1000.98, sqrt(2) etc.

Since you are using the sin^-1 function you will only ever get 1 angle as the range is defined from -90 to 90 degrees(which is -pi/2 to pi/2 in radians). You can it sketch on desmos to see what it look likes.

So if you need to find an obtuse angle then you need to use 180-theta to find the obtuse angle.

It might be helpful to take look at videos related function as well.

Question 2

in problem 3.3 I had to open the explanation and do not understand why the law of sines in this solution is switched to sin(c)/length of side. isnt it usually the other way around??

Accepted Answer

Hello Candacemhazelwood,

The law of sines states:

sin(a)/A = sin(b)/B = sin(c)/C
 
when angle a is opposite side a
when angle b is opposite side B
and
when angle c is opposite side c

Hope this helps:
https://www.khanacademy.org/math/geometry-home/right-triangles-topic/law-of-sines-geo/v/law-of-sines  solve for side

https://www.khanacademy.org/math/geometry-home/right-triangles-topic/law-of-sines-geo/v/law-of-sines-example - solve for angle

Question 3

So, obviously, there is the law of sines and the law of cosines. That is what this entire section has been about. However, I'm curious about if there is such a thing as the law of tangents. Since there is both sine and cosine, wouldn't it make sense if there was something like the law of tangents?

Accepted Answer

Yes there is. Though I will admit that the only way I know that is by looking it up. I assumed that was but wasn't certain. You can probably find the exact statement of the law on Wikipedia or some math site.

Question 4

I can't find anything here about ambiguous triangles.  What if a question asks you to solve from a description where two triangles exist? Like "Determine the unknown side and angles in each triangle, if two solutions are possible, give both: In triangle ABC, <C = 31, a = 5.6, and c = 3.9."  I solved for height and see that two solutions exist, and the answer key in my textbook agrees, but I can't figure out how to get either.  From a set of questions that's only supposed to be on sine law.

Accepted Answer

Use the Law of Sines to get one possible angle A:
sin(A)/a=sin(C)/c
sin(A)/5.6=sin(31)/3.9
sin(A)=5.6sin(31)/3.9
A=arcsin(5.6sin(31)/3.9)=47.6924

Subtract 31 (C) and this angle (A) from 180 to find the third angle (B=101.3076) and use the Law of Sines again to find the third side. If you use the given angle-side pair (C and c) you will be less likely to incur error from your own rounding of angle A:
b/sin(B)=c/sin(C)
b/sin(101.3076)=3.9/sin(31)
b=3.9sin(101.3076)/sin(31)=7.4253

But if you know that supplementary angles share a sine value, you know that A can also be an obtuse angle with the same sine as 47.6924:
A=180-47.6924=132.3076

And again, subtract 31 (C) and the obtuse angle A from 180 to find the other possible third angle (B=16.6924) and use the Law of Sines to find the other possible third side, again using angle C and side c to avoid errors from rounding:
b/sin(B)=c/sin(C)
b/sin(16.6924)=3.9/sin(31)
b=3.9sin(16.6924)/sin(31)=2.1750

It all comes from knowing that there are two angles, one obtuse and one acute, for every sine value. And you find the obtuse one by subtracting the acute one from 180.

If you try to do this with a unique triangle (one without two possible sets of angles and sides) your given angle and the obtuse angle you find will add up to more than 180, and so if you try to find a third angle to go with the obtuse one, your subtraction will tell you the third angle is negative, at which point you know you're going down a nonsensical mathematical road, and there weren't two possible triangles to begin with.

Good luck!

Question 5

In the Video 'Solving An Angle With The Law Of Sines' at 2:05, Sal said that Law of Sines is 'sin(a)/A = sin(b)/B = sin(c)/C (lower case letters are the angles and the upper case letters are the side opposite to the angle), in this article it says 'A/sin(a) = B/sin(b) = C/sin(c)'. So which one should I choose?

Accepted Answer

The Law of Sines can be written either way! You can put the angles in the numerators and the sides in the denominators, or the other way around.

To understand why, think about this true equation:

1/2 = 2/4 = 3/6

It's still true if we reverse the numerators and denominators:

2/1 = 4/2 = 6/3

Question 6

Is this a correct way to check whether you use the law of sines or law of cosines? If you have two angles and a side or two sides and an angle, use the law of sines, if you have 3 sides use the law of cosines. Is there anything you need to add to this list?

Accepted Answer

The law of sines works only if you know an angle, a side opposite it, and some other piece of information. If you know two sides and the angle between them, the law of sines won't help you.

In any other case, you need the law of cosines.

Question 7

If law of sines is a/sin(a)=b/sin(b)=c/sin(c), does sin(a)/a=sin(b)/b=sin(c)/c work?

Accepted Answer

Yes. Just raise one of those equations to the -1 power, and you get the other equation. They're equivalent.

Question 8

What is the numeric form of sine, cosine, and tangent? Like what does the calculator use to calculate trig functions?

Accepted Answer

The trig functions can be expressed as polynomials of infinite degree, called Taylor polynomials. For example, sin(x)=x-x³/6 +x⁵/120 -x⁷/5040+... Some calculators use these series to approximate the values of the trig functions, as well as other functions like exponentials.

You'll develop Taylor polynomials in calculus.

You'll develop Taylor polynomials in calculus.

Question 9

What is the law of tangents used for??

Accepted Answer

The law of tangents is used to keep people from getting to the point. Old people tend to use it a lot :)

Course: Precalculus > Unit 2

Laws of sines and cosines review

Law of sines

Law of cosines

Practice set 1: Solving triangles using the law of sines

Example 1: Finding a missing side

Example 2: Finding a missing angle

Practice set 2: Solving triangles using the law of cosines

Example 1: Finding an angle

Example 2: Finding a missing side

Practice set 3: General triangle word problems

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