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### Different Answers with Sine Rule and Cosine Rule

```Date: 08/07/2002 at 02:39:12
From: Thomas Saw
Subject: Different answers with Sine Rule and Cosine Rule

This is really strange. I found this problem in a textbook. A triangle
ABC has measurements AB = 8.2cm, BC = 9.4cm and AC = 12.8cm and
angle A = 47 degrees. Find angle B.

Method 1 - Using the Sine Rule;
sin B = AC (sin A)/BC gives B as 84.4 degrees.

Method 2 - Using the Cosine Rule;
AC^2 = BC^2 + AB^2 - 2(BC)(AB) cos B gives B as 93.1 degrees, which

Why does using the Sine Rule here fail to give the right answer?

```

```
Date: 08/07/2002 at 09:19:58
From: Doctor Rick
Subject: Re: Different answers with Sine Rule and Cosine Rule

Hi, Thomas.

It's strange, isn't it? There are several things going on here at the
same time, making it a bit hard to untangle.

First, the triangle is overspecified to begin with. Given just the
lengths of the sides, you can use the law of cosines to find angle A.
It turns out to be, not 47 degrees exactly, but 47.165 degrees.

Second, since angle B is close to 90 degrees, a small error in the
sine of the angle can result in a rather large error in the angle
itself. In particular, when I apply the law of sines with 47 degrees
for A, I get sin(B) = 0.99588589; but when I use 47.165 degrees for
A, I get sin(B) = 0.99857047. That's a small error - but when I take
the inverse sine of each value, I get 84.80 (I don't know why this is

Now look at the corrected answer of 86.936 degrees using the law of
sines. Compare it with the correct answer found using the law of
cosines: 93.064 degrees. Each is 3.064 degrees away from a right angle
- but in opposite directions! Does this give you an idea of
what's going on?

When you use the law of cosines to find angle B, you are making use
of the three sides of the triangle. Remember the SSS congruency
theorem: three sides determine the shape of a triangle uniquely. This
is why we are able to find a unique measure for angle B.

When you use the law of sines, you are making use of two sides and an
angle. Notice how these are related: the angle is NOT between the
sides. In terms of congruency theorems, we need an SSA theorem - but
there isn't one. The triangle is NOT uniquely determined by these
quantities.

Remember that we found that the sine of angle B is 0.99857047. We
can't just take the inverse sine of this number: that is ONE angle
with this sine, but 180 degrees minus this angle is another solution.
The latter solution turns out to be correct in this case.

You see, there are two triangles that have the same angle A and sides
AC and BC. The law of sines, applied without thinking, gave one of
these triangles, but it was the wrong one. You must be careful in
taking the inverse sine: note both solutions in the range 0 to 180
degrees, and check whether each leads to a valid solution of the
triangle. If both do, you need more information (such as side AB) to
tell which is correct.

See the Dr. Math FAQ on solving triangles:

Trigonometry Formulas (select Solving Oblique Triangles)
http://mathforum.org/dr.math/faq/formulas/faq.trig.html

Also see these items in the Dr. Math Archives:

Ambiguous Cases - Laws of Cosines and Sines
http://mathforum.org/library/drmath/view/54143.html

Angle-Side-Side Does Not Work
http://mathforum.org/library/drmath/view/54663.html

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Triangles and Other Polygons
High School Trigonometry

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