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Warning: you may not have the correct answer. Second, if you know two sides and the angle opposite one of them, then you can almost determine the angle opposite the other one of them.
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Like the law of cosines, you can use the law of sines in two ways.įirst, if you know two angles and the side opposite one of them, then you can determine the side opposite the other one of them. Note that the law of sines says that three ratios are equal. The law of sines is also a simply stated equation This is fine, and your calculator will compute the arccosine properly. As you can see in the graphs on the previous page, the cosine of an obtuse angle is negative. Suppose the three sides are a = 5, b = 6, and c = 10. Note: When triangle is obtuse, the cos C is negative. Second, if you know all three sides of a triangle, then you can use it to find any angle. Since the cosine of 60° is 1/2, that equation simplifies to c 2 = 49, so c = 7. You can use it in a couple of ways.įirst, if you know one angle and the two adjacent sides, then you can determine the opposite side. The law of cosines relates the three sides of the triangle to one of the angles. Note that each triangle gives three equations for the law of cosines since you can permute the letters as you like. It looks like the Pythagorean theorem except for the last term, and if C happens to be a right angle, that last term disappears (since the cosine of 90° is 0), so the law of cosines is actually a generalization of the Pythagorean theorem. These are called the “law of cosines” and the “law of sines.” There are other “laws” that used to be used, but since the common use of calculators, these two laws are enough. The trigonometry of oblique triangles is not as simple of that of right triangles, but there are two theorems of geometry that give useful laws of trigonometry. Let the angles be labelled A, B, and C, and let the sides opposite them be labelled a, b, and c, respectively. Samuel Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.Let’s agree to a convention for labelling the parts of oblique triangles generalizing the convention for right triangles. If you are my student, please do not contact me here. Please check back for the remaining ones later.Ĭomments, ideas, areas of improvement, questions, and constructive criticisms are welcome. The calculators should work.Īt the moment, only the calculators for the Translations, Reflections, and Rotations are completed. I wrote the codes for these calculators using Javascript, a client-side scripting language. Then use the calculators to check your answers. You are encouraged to: solve the questions graphically (by construction) verify the solutions algebraically (by formulas) Third: solve the questions/solved examples.įourth: check your solutions with my thoroughly-explained solutions.įifth: check your answers with the calculators as applicable.
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