MIT grad shows how to use the Law of Sines to solve a triangle for missing sides and angles. To skip ahead: 1) AAS or ASA case, when you're given TWO ANGLES AND A SIDE, skip to 0:17. 2) For how to know WHEN to use the Law of Sines in trigonometry, skip to 5:55. 3) For when the SSA ambiguous case comes up, and there may be one solution, two solutions, or no solution, skip to 6:32. Nancy formerly of MathBFF explains the trig steps to find the missing sides of a triangle using the sine function in this basic introduction to the Law of Sines formula used in trigonometry, geometry, algebra 2, and precalculus. This trig law is also known as the Sine Rule. It is helpful for solving OBLIQUE triangles with trigonometric functions. For help with and examples of solving a RIGHT triangle using trig functions, jump to https://youtu.be/a5WQlcFTXyk. For an intro to TRIGONOMETRY BASICS (sin, cos, tan, csc, sec, and cot) and SOH-CAH-TOA, jump to: https://youtu.be/bSM7RNSbWhM.
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If you need to "solve the triangle" in a trigonometry problem, it just means to find all the missing angles and sides of the triangle.
WHAT is the LAW of SINES? Say that you're given an oblique triangle, meaning one that is not a right angle triangle, and you need to solve it. If you know two angles and at least one of the sides, you can solve with the Law of Sines trigonometry formula. The Law of Sines says that the sine of one angle over the side opposite that angle, equals the sine of the next angle over its opposite side, which equals sine of the last angle over its opposite length. These three ratios equal each other, and you can use these proportions to solve for unknowns.
Say that you're given that angle B is 34 degrees, angle C is 110 degrees, and side b is 14. We know two angles and one side. What's missing, or what we're looking for, is side a, side c, and the measure of angle A. That missing third angle, angle A, we can find very quickly without using any special new law or trig identity, since the sum of the degrees in a triangle is always 180 degrees. Subtracting the two known angles (34 and 110 degrees) from the total 180 degrees gives that the measure of angle A is 36 degrees.
HOW to use the Law of Sines: to find the missing two side lengths, we do need the Law of Sines theorem. First write all the ratios from the Law of Sines: the sine of each angle, over the length opposite. We write the sine of 36 degrees over the opposite side, side a. This equals sine of the next angle, 34 degrees, over its opposite side 14. And finally, that equals sine of the last angle, 110 degrees, over the opposite length, the unknown c.
If we look at just the leftmost two ratios, we see that we can use that equation to solve for a. So we separate that part out to solve: sin(36)/a = sin(34)/14. An easy way to solve a proportion like this is to cross-multiply. This gives us 14sin(36) = a*sin(34). Now it's more clear how to solve for side a, by just dividing out sin(34) from both sides. So we get that a is equal to 14*sin(36)/sin(34). To get a number value that's practical as the length of a triangle side, you can use your calculator to get a decimal number. CAUTION: since we were given angles in degrees, make sure your calculator is in degree mode, not radian mode, so that you don't get the wrong answer. The side length a is then approximately 14.72.
Now there's just one more unknown to find, the side c. To find the remaining missing side of a triangle, you can use a different pair of ratios to solve, the equation with the rightmost two ratios: sin(34)/14 = sin(110)/c. In general, to solve for one of the unknowns, use the ratio that has what you want to find in it, as the only unknown, and set it equal to a ratio where you know everything already, and you will get the answer. We cross-multiply to get c*sin(34) = 14sin(110). When we get c alone and use a calculator, we find that side length c is approx. 23.53. Now we've completely solved the triangle for all the missing sides and angles.
How do you know WHEN to use the LAW of SINES? If you're given two angles and a side (AAS or ASA cases), or two sides and an angle opposite one of those sides (SSA), you can use the Law of Sines property. If you have two angles and one side (AAS/ASA), you can use the Law of Sines just as we did to find the missing sides. WARNING: for the SSA case, when you have two sides and an angle that's opposite one of them, you can also use the Law of Sines to solve, but instead of having one solution, there may be no solution or two solutions. The SSA case is also called the "ambiguous case".
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Editor: Miriam Nielsen of zentouro @zentouro