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Tm <5E313220> Tj 1 0 0 1 325.62 139.68 Tm <32> Tj 1.50 Tc 1 0 0 1 79.38 111.42 Tm <54616E20> Tj 1 0 0 1 100.98 111.24 Tm <34356F20> Tj 1 0 0 1 119.70 113.76 Tm <3A51723A6C> Tj 1.08 Tc /F3 15.0 Tf 1 0 0 1 237.42 109.80 Tm <636F7420> Tj 1 0 0 1 254.34 109.80 Tm <34356F20> Tj 1 0 0 1 273.24 112.14 Tm <3A2320> Tj 1 0 0 1 309.06 113.76 Tm <2D31> Tj ET endstream endobj 4 0 obj 8052 endobj 5 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 6 0 obj << /Length 7 0 R >> stream Trigonometry: Properties, Special Angles & Identities The angle e is defined by e :1, where s is the distance along therarc, and r is the distance from the center to the arc. This numberis in radians. To convert to degrees, multiply UV-1-99.7r Sine of e : sin o: Cosine of e - cos e Tangent of e : tan e : v4 The six trigonometric identities are all ratios of the threesides of the triangle. They are all definitions and should bememorized cosecantefe-csce- Bv sin I1 secant ofo: sec o: cosd cotangent of e tan0 xRvx 8_x1 _x_Y There are sometrig. Functions triangles fromfor these angles geometry with special angles, and the values of the sixshould be memorized. The first triangle is the 45"-45"-90o. 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stream Trigonometry: Properties,The other triangle is the 30o-60o-90o Special Angles & Identities Suppose we wanted to find an arigle other than the ones we have memorized: There aretools in trigonometrythat allow us to do this. The first is the idea of coterminal angles.Because of the way the unit circle works, any angle can be represented the exact sameway by adding or subtracting 360o (or 2r in radians). Here is a graph to help see why. 0+znl The first angle is the small one, represented by 0.The larger angle goes all the way around and comesback to the exact same angle, as the straight lineshows. 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These formulas allow us tofind the sin, cosine, or tangent of the sum of two angles. The formulas for sine, cosine,.and tangent are:o sin(A+B) - sin(A)*cos(B) + sin(B)*cos(A)o cos(A+B) - cos(A)cos(B) - sin(A)sin(B)o tan(A+B) - tan('4) + tan(B)1 - tan(A) tan(B) Angle subtraction formulas can be obtained by substituting (-B) for (B) and evaluatingthe formulas. As an example, cos(75") : cos(3 0o+45o) - cos(30")coe(45") - sin(30")sin(45"):(f )(Q) (1 )(Q):Ju-^[,2 2 "2" 2' 4Almost all of the other trig identities can be derived from clever substitutions into these threebasic formulas. For example, the double angle identities can be obtained by setting B equal to A. o sin(2A) - sin(A+A) : sin(A)cos(A)+sin(A)cos(A) - 2sin(A)cos(A)Similarily,o cos(2A) - rort(A) - sin2(A) , and by the pythagorean theorem,o cos(2A) - zcosz(A) - 1 : 1- zsin2(A). tan(2A) - 2tan(A)| - tanz (A) endstream endobj 25 0 obj 991 endobj 26 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 27 0 obj << /Length 45 >> stream q 612.18 0 0 792.00 0.00 0.00 cm /Obj21 Do Q endstream endobj 28 0 obj << /Type /Page /MediaBox [ 0 0 612.18 792.00 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj21 21 0 R >> /Subtype 24 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 26 0 R 27 0 R 22 0 R ] >> endobj 29 0 obj << /Type /XObject /Subtype /Image /Width 3401 /Height 4400 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /Columns 3401 >> /Length 25146 >> stream >?d^? >̅X )gpnp~!"C# P̄0vefaA2fy3=A]2h0ຢ"Y0GYNPRVfh'. 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They can be used toderive the half-angle formulas.If cos(2A) : 1-2sin2(A), then Zsin2(e; - 1- cos(2A), and so o sin(A) - This becomes particularly useful when we make a small substitution in the angles. If we let B:2A,a then l: !, and so _,-, B ,, _ /1- cos(B)sm( - l: .,-'2' tl Z Similar work will eive us,8, 11+cosB. cos( 1): \ 2 ,B.. tan(-): +'2' l1- cosB sur,Bt-:_1t+cosB 1+cosB 1- cos.B sin ^B Clever combinations and substitutions of these and other trigonon;etric identities will yreld evenmore identities, such as the sum-to-product and product-to-sum icientities. For a more completelist of trig identities, see our other handout available on the back wall or ask a tutor. 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