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Trigonometry: Reference Angles

The repetitions patterns the trigonometic frrnctions and of makemanycalculations easier. Instead of memorizing manyanglesandthe valuesof the trigonometric functionsfor eachof theangles, can we usethe ideaof reference angles cut downon thenumberof angles needto look at. Reference to we angles definedasthe acute(lessthan90') anglethat a glveq anglemakeswith thex:CI<is are (positiveor negative). thesegraphs show,the referenceanglein the first quadrant is itself; in the Jecond As quadranf is what is missingfrom one-halfrevolution;in the third quadranf it is what is extra from one-halfrevolutionl and in the fourth quadran! it is what is missingfrom onewhole revolution.

ln eachof these four graphs, outerarc is the original angle,andthe inner arc is thereference the angle. Wheredoesthis get us?Themajorvalueof reference angles comesfrom the fact thatfor any trigonometric function,we canexchange angleandits reference an angle,andthe answer exactlythe is same, long aswe give it the correctsign.How do we know which sign to give it? That comesfrtm as knowingwherethe functionsarepositiveandwherethey arenegative. This canbe remembered easily by usingthemnemonic "All Students Take Calculgs"-sfs{ingin the first quadrant, countergo clockwise,assigning eachletterto eachquadrant. You canalsousethe word CAST.Using CAST, start inthefourth quadrant. Eitherway, we have: A stands "all", which means for that all the functionsarepositivein the first quadrant. stands "sine", which means S fbr that only sineis positivein the second quadrant. is for "tangent",andC is for "cosine". T Numerically, way to find a reference the anglefor some angle,d, is as follows: in quadrant it is the same 0. In quadrant it is equallo lg0 - d. kr I, as II, quadrant it is d -180.Finally, in quadrant it is 360 - d. III, fV, So, asan example, let's look at finding tangent135'. This is not a memorized value,so let,s try using reference angles. 135' is in thesecond quadrant, our reference ", so angleis 180'-135 or 45'. We know that the tangent 45' is 1, but is it positiveor negative? of This is wherewe usethe CAST rule. In quadrant two, only sineis positive,so tangent negative. is Therefore, (135') = -1. tan Reference angles alsocomeinto playwhen we areevaluating your inversetrigonometric functions. calculator may give you an answer these, it is not alwaysthe correit answer. to but This is because for a functionto havean inverse, mustbe one-to-one, someapplications trigonometry not it but of are limited in this way. So,you may be giventhe following: sin $); -\[2/Z and asked find x, wherex is to in the third quadrant. = Your calculator will tell you that sinl 1-.r12tz) -45', but this is not in the third quadrant. Because calculator the gaveus -45",we know that the reference angleis 45', sowe needto find what anglein the third quadrant a reference has angleof 45'. Using the rulesofreferenceangles, we seethat it is 225'.If we checkon the calculator, (225') is in fact-{2/2. sin


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