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In my math class (precalc) my book states the following rules for solving an SAS triangle using law of cosines.

Solving an SAS Triangle

  1. Use the Law of Cosines to find the side opposite the given angle.

  2. Use the Law of Sines to find the angle opposite the shorter of the two given sides. This angle is always acute.

  3. Find the third angle by subtracting the measure of the given angle and the angle found in step 2 from 180.

Why do you need to follow step #2?

N. F. Taussig
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wiebecoding
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    You could also use the Law of Cosines to find that angle, so I guess it's just for the practice of using the Law of Sines. – Paul Jan 09 '20 at 16:53
  • I've revised my answer. I tried to be make it clear and detailed. If you have additional questions, please let me know. I'll be happy to help. – user729424 Jan 09 '20 at 17:27
  • One can also use $$ \cos(\mathrm{A})=\frac{b-a\cos(\Gamma)}{\sqrt{a^2+b^2-2ab\cos(\Gamma)}} $$ which, since $0\lt\mathrm{A}\lt\pi$, gives $\mathrm{A}$ unambiguously. – robjohn Jan 10 '20 at 10:30

3 Answers3

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Let's say that we've completed Step #1, and we're trying to decide what to do next. At this point, we know all the sides of the triangle, we know one of the angles, and we're trying to find the other angles. Let's suppose $\theta$ is one of the angles, and we're trying to find $\theta$.

When you use the law of cosines to find some angle $\theta$ of a triangle, first you try to get $\cos\theta$ by itself, and you end up with an equation of the form

$$\cos\theta=\text{BLAH}$$

($\text{BLAH}$ is just some number). Since $\theta$ is the angle of triangle, you know that $0<\theta<180^{\circ}$, so you get that

$$\theta=\cos^{-1}\text{BLAH}.$$

Now the formula for the law of sines is a bit less complicated than the formula for the law of cosines, so perhaps it would be better to use the law of sines. If we use the law of sines, there will be another issue that we have to worry about, as we will see below.

When you use the law of sines to find some angle $\theta$ of a triangle, first you try to get $\sin\theta$ by itself, and you end up with an equation of the form

$$\sin\theta=\text{BLAH}$$

(Again, $\text{BLAH}$ is just some number). And again, we have that $0<\theta<180^{\circ}$, since $\theta$ is an angle of triangle. Note that the sine function is positive in quadrants one and two. Hence there are two possibilities:

$$\theta=\sin^{-1}\text{BLAH}$$

$$\theta=180^{\circ}-\sin^{-1}\text{BLAH}.$$

One of these gives the correct answer for $\theta$. Which one is it?

Your book has a clever way around this issue. Of the two given sides, the angle opposite the shorter given side will be shorter than the angle opposite the other given side. Hence the angle opposite the shorter given side will not be the largest angle of the triangle. So it must be acute. So if $\theta$ is the angle opposite the shorter given side, then $0<\theta<90^{\circ}$. Hence, we'll have just one answer for $\theta$:

$$\theta=\sin^{-1}\text{BLAH}.$$

So we should first try to find the angle that's opposite the shorter given side. That way, we can use law of sines (which is easier than law of cosines), and we won't have to worry about getting more than one answer.

user729424
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You don't have to follow Step 2, but it helps.

To start with, you have no choice but to follow Step 1. The Sine Rule would be no help to you in that situation.

Having followed Step 1, you now have three sides and one angle. You could either use the Cosine Rule to find one of the unknown angles or you could use the Sine Rule. The Sine Rule is generally perceived as easier - there are fewer operations to perform, perhaps...

Anyway, having decided to opt for the Sine Rule as your method, we now have the possibility that you could have an ambiguous case, where it is unclear whether the angle you are finding should be acute or obtuse. Choosing to find the angle opposite the shorter of the two original sides guarantees that your angle is either the smallest angle in the triangle or is the second smallest (which might be the case if the original angle is the smallest angle). What's clever about this choice is that there must be at least one angle greater than the angle you are finding, which guarantees that your angle can not be obtuse. So no ambiguity when using inverse Sine.

To avoid ambiguity you could instead have Step 2: Use Cosine Rule to find one of the other angles. No ambiguity there.

tomi
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It's nice to have a clear algorithm to solve a given triangle, but by doing so, it doesn't encourage creativity. For example, the book could say that "To solve an SAS triangle you will need to find another angle or side. Once that is known, the law of sines can be used to find a missing side or angle."

If you take step #1, then the wisdom of calculating the angle opposite the smaller side has been discussed above. But you could also use the law of tangents (or a variant) to solve for the smaller angle straight away (without calculating the other side): given $a,b,\angle C$ with $a\le b$ calculate $\angle A$ from $$tan(A) = \frac{sin(C)}{b/a-cos(C)}$$ Next, you can calculate the missing side using the law of sines (since you now have an angle-side pair). Finally, the angle sum can be used to find the remaining angle.

You could save the law of cosines for solving the SSS triangle.

smichr
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