Solving Triangles

    The basics idea in solving triangles is that you are given some information about a triangle and then the must find the missing information. In what follows we will find the missing information in the triangle:



where is the angle opposite side of length is the angle opposite the side of length and is the angle opposite the side of length

    Recall, the law of cosines states







and the law of sines states,



Also recall that just because you are given information about a triangle, that does not mean there is a triangle meeting such a description.  Here is a summary:



Note that when an angle is found by means of the law of cosines, there is no ambiguous case since we always obtain a unique angle between and

Example (Solving Triangles) If then find To find we will use the formula and with substitution we have







Therefore, the value for is

Example (Solving Triangles) If and then find To find we will use the formula and with substitution we have







Therefore, the value for is Notice that in the previous example the angle was acute and in this example the angle is obtuse.

Example (Solving Triangles) If then find in radians. To find we use the formula and with substitution we have









Therefore we find that angle is In this case there is only one solution because the cosine function is negative in quadrants 2 and 3. But only quadrant 2 has angles that are less than Note also that any triangle is determined given all three sides.

Example (Solving Triangles) If then find in radians. To find we use the formula and with substitution we have









In this case, there is only one solution for because the cosines function is positive in quadrants 1 and 4. But only quadrant 1 has angles that are less than Therefore, we find angle to be Note also that any triangle is determined given all three sides.

Example (Solving Triangles) If then find in radians. To find we use the formula and with substitution we have









In this case, there is only one solution for because the cosines function is positive in quadrants 1 and 4. But only quadrant 1 has angles that are less than Therefore, we find angle to be Note also that any triangle is determined given all three sides.

Example (Solving Triangles) A triangle has two sides of length 11 cm and 24 cm with a angle adjacent to the 24-cm side. Find the other side and angles.

    Solution. Let We can find using the law of sines:



and so Since for any angle no such triangle can exist.

Example (Solving Triangles) If a triangle has sides of 15.0 ft and 25.0 ft and a angle adjacent to the 25.0-ft side, find all possible solutions for this triangle.

    Solutions. Let and We can use the law of sines to find



So Therefore, there are two values for angle Namely and Case 1 is when and in this case we find

    

and so



Therefore the length for is ft. Case 2 is when and in this case we find

    

and so



Therefore the length for is ft.

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Cite this as:
Solving Triangles
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/solving-triangles.html