Trigonometry Notes: Solving Triangles

The three angles must add up to

or else there is no such triangle. If the three angles do add up to

then there are infinitely many triangles with those three angles. Three angles does not determine a triangle, because you can rescale the three sides by the same factor without changing the angles. For instance there is a triange with angles

whose side lengths are

, but there is also one with side lengths

, or

, etc.

For any SAA there is one triangle with the given information if the two angles add up to less than

, and no triangles if the two angles add up to greater than

. If you know two of the angles then you can find the third, since the three angles should add up to

. Then you can use the Law of Sines to find the other two side lengths.

Suppose, for example, that you know , , .

Then, since , we also know that . From the Law of Sines:

Similarly,

For any ASA there is one triangle with the given information, provided the two angles add u pto less than

. If you know two of the angles then you can find the third, since the three angles should add up to

. Then you can use the Law of Sines to find the other two side lengths, just like the SAA case.

This is the tricky one. There may be zero, one, or two triangles with the given information. Assume the given information is angle

, and sides

and

Don't try to memorize all of the following cases, try to visualize what's happening in each case. Imagine you are trying to make a triangle out of sticks. You know how long sticks and are. We don't know how long the third stick should be, but we can imagine that it will lie flat on the ground. We know that stick should come out of the ground at angle . We don't know what the angle between sides and should be, so imagine they are connected by a hinge. We want to know how we can set this up so that the other end of stick touches the ground.

Draw side coming out of the ground at angle . Draw of circle of radius centered at the end of side . Side must have one endpoint at the center of this circle and the other endpoint a a point where the circle intersects side .

For any triangle we find the procedure will be the same. We know sides and and angle , so use the Law of Sines to find angle .

There are two possible solutions for , either or . To figure out which one to use you must look at your picture. is an acute angle, while, is an obtuse angle, so pick the one which maches the picture.

Now we know and , so , and finally we can use the Law of Sines to find

is acute
Let .
- No triangles. Side does not reach the ground, so the triangle can't close up.
- One triangle. Side can reach the ground in exactly one spot.
  
  In this case we get a right triangle, .
- Two triangles. Side can reach the ground in two spots.
  
  In this triangle is an acute angle, so use
  
  In this triangle is obtuse, so use
- One triangle. Side can reach the ground in two spots, but one of them doesn't correspond to a triangle with angle .
  
  There is only one triangle to consider, and is acute, so use .
- No triangles. Side doesn't reach the ground on the correct side of side .
- One triangle. Side can reach the ground in two spots, but one of them doesn't correspond to a triangle with angle .
  
  In this triangle is an acute angle, so use

For any SAS there is exactly one triangle with the given information, assuming the angle is less than

Let's suppose, for example, that we know , and

Connect the two open ends to make a triangle.

Now use the Law of Cosines to find side .

Now we have all three sides, so see SSS.

There is either one or zero triangles with the given side lengths. The triangle inequality tells us that the longest side should be shorter than the sum of the other two sides. If this is true there is one triangle with the given side length, if it is false there are no such triangles.

Given three side lengths that satisfy the triangle inequality you can find the three angles by using the three versions of the Law of Cosines.

Suppose we have side lengths , , and .

There is a triangle with these three side lengths, since is the longest side and . You could show this with a sketch by drawing a segment of length 15 for he longest side. Then draw circles centered at each edge of the segment with radii equal to and and see that the two circles intersect. The point where they intersect is the third vertex of the triangle:

So we have a triangle

One method would be to use the Law of Cosines three times to find the three angles. To find angle $C$:

Similarly:

You could have also used the Law of Cosines twice and then subtracted the two agles form to get the third.

Another method would be to use the Law of Cosines once and then use the Law of Sines twice to find the other two angles, or use the Law of Sines once and then subtract the two angles from to get the third.

When using the Law of Sines you must be careful to consider whether you need or . You can avoid this problem by always doing the Law of Cosines in the first step to find the largest angle. Then the other two angles must be acute, so you only need to take in you solutions.

For example, if you found with the Law of Cosines then you could ust the Law of Sines to find:

and

But if you found with the Law of Cosines then you need:

and