Two special Righttriangles 1. The


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1 Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The right tringle. We cn construct right tringle with 45 ngle. The tringle hs two 45 ngles. Therefore, the tringle is isosceles tht is, it hs two sides of the sme length. ssume tht ech leg of the tringle hs length. We cn find the length of the using Pythgoren s Theorem. 45
2 . The right tringle. There re two other ngles tht occur frequently in trigonometry, 0 nd 60. We cn find the vlues of the trigonometric functions for these ngles using right tringle. To form this right tringle, drw n equilterl tringletht is tringle with ll sides the sme length. ssume tht ech side hs length equl to. If we drw line right down the middle of this tringle isecting the top ngle nd dividing the se into two equl prts, then we will hve right tringle. See the figure elow. 0 We cn find the length of the missing side,, using Pythgoren s Theorem
3 B (length of )  c  (length of side ) (Length of side to ) The definitions of the six trigonometric functions of the cute ngle re s follows: length of side c length of sin csc c length of length of side length of side to c length of cos sec c length of length of side to length of side length of side to tn cot length of side to length of side Exmple: Find the vlue of ech of the six trigonometric functions of in the figure elow. B c = = 4 Solution: In order to evlute ll six trigonometric functions, we need to know the length of ll sides of the tringle. Since the lengths for sides nd re given, we cn use Pythgoren s Theorem, c = +, to find the length of side c. c sin = csc = c c cos = sec = c tn = cot =
4 45 Now tht we know the lengths of the sides of the right tringle, we cn find the six trigonometric function vlues for the ngle = 45. sin 45 csc 45 cos 45 sec 45 tn 45 cot 45 Now tht we know the lengths of the sides of the right tringle, we cn find the six trigonometric function vlues for oth ngles = 0 nd = 60. Using this right tringle to fill in the lnks elow. 0 sin 0 csc0 cos0 sec0 60 tn 0 cot 0 = = sin 60 csc 60 cos 60 sec 60 tn 60 cot 60 = =
5 The trigonometric function vlues for n ngle depend only on the size of the ngle, nd NOT on the size of the tringle. = 6 = =.5 = = 4 = = = 4.5 Notice tht ll of these right tringles hve the sme ngle,. Even though the tringles re different sizes, they re similr. This mens tht the tringles hve the sme shpe nd the lengths of the corresponding sides re in the sme rtio. Becuse n cute ngle in right tringle lwys gives the sme rtio of to sides, the trigonometric functions vlues evluted for the ngle will e the sme for ll of these tringles. Since we know tht the size of the tringle is not importnt, it is helpful to look t right tringles in which the length of the is equl to. Here re our specil tringles djusted so tht the length of the is. The Bsic Right Tringle The Bsic Right Tringle 0 45 = Fcts: For ny the length of the shortest leg is lwys = times the length of the. For ny times the length of the shortest leg. For ny times the length of leg.
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