# Operations Involving Law of Sines and multiple Unknown Values

• Liquid7800
In summary: Yes, this is a reasonable solution. The Law of Cosines is more efficient because it only requires the input of one vector (the length of the other two vectors).Thanks all!In summary, the problem is trying to figure out the length of |b| when |a| = 6 and b = 30. The Law of Sines can be used to solve this problem.
Liquid7800
Hello, I was working on a computer graphics problem when I encountered an interesting sceanrio:

I have two vectors a and b, such that the angle between them is 45 degrees.

The vector a+b and a have an angle between them that is 30.

This produces a problem in 'drawing' a triangle, when trying to solve this problem using the law of sines because the 'triangle' involving vectors a, b, and a+b do not 'add up' to 180 degress because we already have two angles (45 and 30) totalling 75 degrees, thereby needing an angle > 90 within this particular triangle for a sum total of 180 degrees for the triangle (since an individual angle of a triangle can't be more than 90 degrees.

In this case, if the length of |a| = 6
then I have one unknown angle (between b and (a+b) and two unknown magnitudes (|b| and |a+b|)

To therefore 'build' a triangle to solve this problem using the Law of Sines, can I simply 'divide' the angle between a and b or a and a+b to create my triangle, to find |b|?

I appreciate any insight of approaching this problem from the angle of using the Law of Sines (no pun intended), and I hope I discribed my problem clearly enough

Let a, b, c be the sides and A, B, C be the opposite angles. c=a+b. I will use | | for side length.

Your description has C=135 (180-45) and B=30. Therefore A=15. Now use the law of sines to get |b| and |c|.

Mathman,

Sigh, thanks...I looked over this problem and didnt see to approach the problem in that simple and elegant manner...thank you...but how did you know to find C= 135 where (180 -45) = 135?...its probably a simple operation too...thanks again

Yes, that's exactly right. Draw a diagram of the vector addition and you will find that the "inside" angle of the triangle is the supplement of the angle between the vectors.

Thanks all you guys...you are correct---and I did that last night. I did do that exact same operation...didnt realize the term was called 'supplement' (I suppose by the paralleogram law?)

In addition, I researched this problem, or type of problems, and is this considered one of the 'Ambigous Triangle Case' types?

I had never heard of 'Ambiguous Case' in relation to triangles, so it was an interesting read.

This is not "Ambiguous case". If you remember the congruent theorems in elementary geometry, they all are based on equality involving three corresponding parts between the two triangles, at least one of which is a side. These are usually described by initials (S for side and A for angle), namely SSS, SAS, ASA, and AAS. Note that SSA is not included, since there are, in general, two different triangles with equality for two sides and an angle (acute) other than the included angle. This is the "ambiguous case".

Hello,

Would any of you be able to verify my 'proof' of a similar situation?

Here it goes:

If the angle between the vectors a and b is 60 degrees and |a| = 6
and the angle between a-b and a is 30 degrees --->

then the |b| can be found as follows:

[1] If vectors a, b, and a-b form a Triangle by vector addition such that the angles of
a + b + (a-c) = 180

[2] then it follows that the angle between b and a-b is 90 degrees

Therefore if |a| = 6 and by using the law of sines we can find |b| by

sin (90)/6 = sin (30)/|b| ...such that sin(A)/|a| = sin(B)/|b| and statement [2]

thus, |b| = 3

Is this a reasonable solution or are any of my assumptions / statements misguided?
In addition, is there any reason why one would need to use the Law of Cosines to figure this out? From a program perspective, this appears to be a much more efficient method (Law of Sines I mean)
Thanks for any help.

a + b + (a-c) = 180
Slight error - I presume the 'c' should be 'b'.

It looks OK otherwise.

## 1. What is the Law of Sines?

The Law of Sines is a trigonometric law that relates the side lengths of a triangle to its angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles of a triangle.

## 2. How is the Law of Sines used in operations involving multiple unknown values?

In operations involving multiple unknown values, the Law of Sines can be used to solve for missing side lengths or angles in a triangle. By setting up a proportion using the given known values and the Law of Sines formula, the missing value can be calculated.

## 3. What are the steps for using the Law of Sines to solve for unknown values?

The first step is to identify the known values, including at least one side length and its opposite angle. Then, use the Law of Sines formula (a/sin A = b/sin B = c/sin C) to set up a proportion. Solve for the missing value using algebraic methods, such as cross-multiplication. Finally, check your answer by plugging it back into the original equation.

## 4. Can the Law of Sines be used for any triangle?

Yes, the Law of Sines can be used for any triangle, regardless of its shape or size. However, it is most commonly used for solving triangles that are not right triangles.

## 5. Are there any limitations to using the Law of Sines?

The Law of Sines can only be used to solve for missing values when there is enough information given, typically at least one side length and its opposite angle. Additionally, the Law of Sines may give ambiguous or no solutions in some cases, such as when trying to solve for an angle opposite a given side length that is longer than the sum of the other two side lengths.

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