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### Finding the Third Side

```
Date: 04/24/99 at 23:43:25
From: Kat
Subject: Trigonometry...

I have forgotten how to find a third side of a scalene triangle using
Trig. I've been given two side measures (5 and 4.2426) and an angle
measure (50 degrees), but I don't remember how to get the third side

Thank you so much,
Kat
```

```
Date: 04/26/99 at 14:55:28
From: Doctor Jeff
Subject: Re: Trigonometry...

Hello, Kat.

The dimensions you gave actually can determine three different scalene
triangles, depending on whether the 50-degree angle is opposite the
side measuring 5, opposite the side measuring 4.2426, or between these
two given sides.

To solve this problem, you need to remember the Law of Cosines.

The law basically states that for a triangle with vertices A, B, and C
and sides a, b, and c opposite their respective vertices, the length
of side c is given by the formula

c^2 = a^2 + b^2 - 2*a*b*cos(C),

where a and b are the lengths of two of the sides and C is the measure
of the angle between them.

This formula should look very familiar. It is the general case of the
Pythagorean Theorem, which states

c^2 = a^2 + b^2,

where c is the length of the hypotenus of a right triangle and a and b
are the lengths of the other two sides. Using the Law of Cosines, we
can see why the Pythagorean Theorem works for a right triangle:

c^2 = a^2 + b^2 - 2*a*b*cos(90)
= a^2 + b^2 - 2*a*b*0
= a^2 + b^2

The Law of Cosines is, therefore, much more powerful than the
Pythagorean Theorem, because it handles every kind of triangle, not
just those with a 90-degree angle in them.

The simplest scalene triangle for which to find the missing side is
the last one listed above. In this case, 50 degrees would be the angle
between the sides of length 5 and 4.2426. You can therefore plug the
numbers directly into the Law of Cosines formula to get your answer.

You might be wondering how to find the missing side for the other two
scalene triangles you can get by moving the 50-degree angle around. In
order to use the Law of Cosines, you must first figure out the measure
of the angle between the sides of length 5 and 4.2426. To do so, you
can use another important trigonometric relation, the Law of Sines. In
the triangle described above, this law states that

sin(A)   sin(B)   sin(C)
------ = ------ = ------
a        b        c

Let's look at the case where the side of length 5 is opposite the
50-degree angle. It would be helpful to draw a picture of this
triangle.

Using the Law of Sines, we can find B, the measure of the angle
opposite the side of length 4.2426.

sin(50)   sin(B)
------- = -------, so
5      4.2426

4.2426 * sin(50)
---------------- = sin(B), so
5

/ 4.2426 * sin(50) \
B = arcsin |  ----------------  |
\        5         /

You now know that the measure of the angle opposite the side of length
5 is 50 degrees and that the angle opposite the side of length 4.2426
is the expression equal to B. Since the sum of the angles of a
triangle is 180 degrees, the angle (C) between the sides of length 5
and 4.2426 is given by

C = 180 degrees - 50 degrees - B

With this angle, you can now use the Law of Cosines to solve for the
other side. Remember that there is yet another scalene triangle that
can be formed with the given constraints; the length of the missing
side can be found in a similar way.

I hope this helped. Good luck with the problem, and don't hesitate to
write back if you still have questions.

- Doctor Jeff, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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