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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 
length. Could you please help?

Thank you so much,

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 

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

                   / 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   
Associated Topics:
High School Trigonometry

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