Honors Geometry Solving Triangles Using The Laws of Sines and Cosines
IMPORTANT NOTE AGAIN: A convention in naming parts of triangles is that capital letters are for angles and small letters are for sides. An angle and the side opposite it have the same letter.
Definitions of three trigonometric functions. oppositesidesin(x)，These definitions only apply to right triangles. hypotenuse
oppositesideXtan(x)，Adjacentadjacentsideside Law of Sines Law of Cosines 222sinAsinBsinCc，a？b？2abcosC ，，;；abc ？1sin(x)Ambiguity with .
？1sin(x)When using always remember that the calculator gives you only one possible angle. The supplement is also a possible angle.
？1sin(0.5)Example: = 30º according to your calculator but sin(150º) = 0.5 also and you must check ;；
both possibilities. In all triangles the largest side must be opposite the largest angle, etc. ;；？1cos(x)There is no such ambiguity with . Every angle from 0º to 180º has a unique cosine and you
don't need to think about two possibilities with this function.;；
Part A. Review how to use the Laws of Sines and Cosines to solve triangles. ;；Sketch each triangle and find all missing sides and angles using the laws of sines and cosines.
1. b = 4, c = 10, A = 75º. In this case you have SAS and will start with the law of cosines. After you find a you will find the remaining angles by the law of sines. Be sure when you are done that
！！you have chosen B and C correctly so that the angles are in the same order of size as the sides opposite them.
2. A = 29º, B = 41º, c = 18. This is ASA and you will find the remaining parts using the law of sines ;；;；alone.
3. A = 100º, a = 12, B = 43º. This is AAS and the law of sines will be all you need.
4. a = 6, b = 12, c = 16. This is SSS. What to do? Use the law of cosines to find any one angle! Then finish up with the law of sines to find the other two. Again be sure using the law of sines that you choose the correct angle so that the largest angle is across from the largest side, etc.
5. b = 22, c = 25, B = 47º. Note for this triangle you are given SSA, which should always be a warning that you are on uncertain territory as to whether there are two possible triangles, or just one, or perhaps none at all! Take a moment and draw two possible triangles in a situation where you are given SSA. In
some real situations you don't know what you have until you try to find the missing parts. Here solve for ！C. You should end up with two complete possible triangles that both fit the given conditions.
6. b = 15, c = 25, B = 47º. SSA again! Do you find one, two or no possible triangles. Explain.
;；7. b = 50, c = 25, B = 47º. SSA again so watch out. Do you find one, two or no possible triangles. Explain.
1. Find all missing sides and angles of these triangles. It may be that the triangle is impossible or it may
be that more than one triangle is possible from the given information. Describe all possible cases fully.
1114X Q ！Q！X
70 D Fe
2. The base of an old factory smokestack is inaccessible and you wish to measure its height from the ground. You make a measurement of its angle of elevation from a certain distance and get 39º. Moving 275 feet further away the angle is 21º. How tall is the smokestack (height off the ground)? Assume the ground is level throughout this figure.
21 39 275 ft
Part C. Bearing problems
1. A ship sails at a bearing of 200º for 75 miles, then sails another 10 miles on a bearing of 340º. At what bearing should it sail to return directly the starting point and how far must it sail?
2. A ship sails north for 50 miles, east for 70 miles and on a bearing of 100º for 125 miles. If a ship were to sail directly to this point from the starting point at what bearing should it sail and how far?