1 of 8 8/6/2004 8.1 THE LAW OF SINES 8.1 THE LAW OF SINES Congrueny and Olique Triangles Derivation of the Law of Sines Appliations Amiguous Case Area of a Triangle Until now, our work with triangles has een limited to right triangles. However, the onepts developed in Chapter 5 an e extended to all triangles. Every triangle has three sides and three angles. If any three of the six measures of a triangle are known (provided at least one measure is the length of a side), then the other three measures an e found. This is alled solving the triangle. Congrueny and Olique Triangles The following axioms from geometry allow us to prove that two triangles are ongruent (that is, their orresponding sides and angles are equal). Congruene Axioms Side-Angle-Side (SAS) If two sides and the inluded angle of one triangle are equal, respetively, to two sides and the inluded angle of a seond triangle, then the triangles are ongruent. Angle-Side-Angle (ASA) If two angles and the inluded side of one triangle are equal, respetively, to two angles and the inluded side of a seond triangle, then the triangles are ongruent. Side-Side-Side (SSS) If three sides of one triangle are equal, respetively, to three sides of a seond triangle, then the triangles are ongruent. Keep in mind that whenever SAS, ASA, or SSS is given, the triangle is unique. We ontinue to lael triangles as we did earlier with right triangles: side a opposite angle A, side opposite angle B, and side opposite angle C. A triangle that is not a right triangle is alled an olique triangle. The measures of the three sides and the three angles of a triangle an e found if at least one side and any other two measures are known. There are four possile ases. Data Required for Solving Olique Triangles Case 1. One side and two angles are known (SAA or ASA). Case 2. Two sides and one angle not inluded etween the two sides are known (SSA). This ase may lead to more than one triangle. Case 3. Two sides and the angle inluded etween the two sides are known (SAS). Case 4. Three sides are known (SSS). 1
2 of 8 8/6/2004 8.1 THE LAW OF SINES Cases 1 and 2, disussed in this setion require the law of sines. Cases 3 and 4, disussed in the next setion, require the law of osines. Derivation of the Law of Sines To derive the law of sines, we start with an olique triangle, suh as the aute triangle elow (figure a) or the otuse triangle elow (figure ). This disussion applies to oth triangles. First onstrut the perpendiular from B to side AC (or its extension). Let h e the length of this perpendiular. Then is the hypotenuse of the right triangle ADB, and a is the hypotenuse of right triangle BDC. In triangle ADB, sin A= h or h= sin A In triangle BDC, sin C = h or h= asin C a h= sin A and h= asin C, Sine asin = sin A a = sin A sin C In a similar way it an e shown that a = and = sin A sin B sin B sin C Thus we have proved the following theorem: Law of Sines In any triangle ABC with sides a,, and, a =, a =, and = sin A sin B sin A sin C sin B sin C This an e written in ompat form as 2
3 of 8 8/6/2004 8.1 THE LAW OF SINES a = = sin A sin B sin C When solving for an angle, we use an alternative form of the law of sines, sin A sin B sin C = = a Note: When using the law of sines, a good strategy is to selet an equation so that the unknown variale is in the numerator and all other variales are known. Appliations If two angles and one side of a triangle are known (Case 1, SSA or ASA), then the law of sines an e used to solve the triangle. Homework exerise 3 4. Find the length of eah side a. Do not use a alulator. Homework exerises 5 16. Determine the remaining sides and angles of eah triangle ABC. 3
4 of 8 8/6/2004 8.1 THE LAW OF SINES Amiguous Case If we are given the lengths of two sides and the angle opposite one of them (Case 2, SSA) it is possile that zero, one, or two suh triangles exist. This situation (SSA) is alled the amiguous ase. # of Possile Triangles -Sketh- Conditions Neessary for Case to Hold 0 sin B > 1, a< h< 1 sin B = 1 a = h< 1 0< sinb < 1, a 4
5 of 8 8/6/2004 8.1 THE LAW OF SINES 2 B2 0< sin < 1, h< a< If angle A is otuse, there are two possile outomes. # of Possile Sketh Conditions Neessary for Case to Hold Triangles 0 sin B 1, a 1 0< sinb < 1, a > We an apply the law of sines to the values of a,, and A and use some asi properties of geometry and trigonometry to determine whih situation applies. The following asi fats should e kept in mind. 1. For any angle θ of a triangle, 0 < sinθ 1. If sin θ=1, then θ=90 and the triangle is a right triangle. 2. sinθ = sin ( 180 θ ) (That is supplementary angles have the same sine value.) 3. The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-valued angle is opposite the intermediate side (assuming the triangle has sides that are all of different lengths). Numer of Triangles Satisfying the Amiguous Case (SSA) Let the sides a and and angle A e given in triangle ABC. (The law of sines an e used to alulate the value of sin B.) 5
6 of 8 8/6/2004 8.1 THE LAW OF SINES 1. If applying the law of sines results in an equation having sin B > 1, then no triangle satisfies the given onditions. 2. If sin B = 1, then one triangle satisfies the given onditions and B = 90 3. If 0< sinb < 1, then either one or two triangles satisfy the given onditions. 1 (a) If sinb=k, then let B1 = sin k and use B1 in the first triangle. () Let B2 = 180 = B1. If A + B2 < 180, then a seond triangle exists. In this ase, use B 2 for B in the seond triangle. Note: The preeding guidelines an e applied whenever two sides and an angle opposite one of the sides are given. Homework exerises 27 32. Find the unknown angles in triangle ABC for eah triangle that exists. Homework exerises 33 40. Solve eah triangle ABC that exists. 6
7 of 8 8/6/2004 8.1 THE LAW OF SINES 46. Property Survey. The surveyor tries again: A seond triangular piee of property has dimensions as shown. This time it turns out that the surveyor did not onsider every possile ase. Use the law of sines to show why. Area of a Triangle The method used to derive the law of sines an also e used to derive a formula to find the area of a triangle. A familiar formula for the area of a triangle is 1 A= h, where A represents area, ase, and h height. This formula annot 2 always e used easily sine in pratie, h is often unknown. Consider the following. 7
8 of 8 8/6/2004 8.1 THE LAW OF SINES h sin A= or h= sin A 1 Sustituting into the formula A= h 2 1 1 A= ( sin A) or A = sin A 2 2 Any other pair of sides and the angle etween them ould have een used. Area of a Triangle (SAS) In any triangle ABC, the area A is given y the following formulas: 1 1 1 A = sin A, A= asin C, and A = asin B 2 2 2 That is, the area is half the produt of the lengths of two sides and the sine of the angle inluded etween them. Note. If the inluded angle measures 90, its sine is 1, and the formula eomes 1 the familiar A= h. 2 Homework exerises 65 69. Find the area of eah triangle ABC. See page 701 for more homework exerises. 8