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Trigonometric Formula SheetDefinition of the Trig FunctionsUnit Circle DefinitionAssume θ can be any angle.Right Triangle DefinitionAssume that:0 θ π2 or 0 θ 90 y(x, y)1hypotenuseoppositeyθxxθadjacentopphypadjcos θ hypopptan θ adjsin θ hypopphypsec θ adjadjcot θ oppy1xcos θ 1ytan θ xcsc θ sin θ 1y1sec θ xxcot θ ycsc θ Domains of the Trig Functionssin θ, θ ( , )csc θ,cos θ, θ ( , )sec θ,tan θ,1 π, where n Z θ 6 n 2 cot θ, θ 6 nπ, where n Z 1 θ 6 n π, where n Z2 θ 6 nπ, where n ZRanges of the Trig Functions 1 sin θ 1 1 cos θ 1 tan θ csc θ 1 and csc θ 1sec θ 1 and sec θ 1 cot θ Periods of the Trig FunctionsThe period of a function is the number, T, such that f (θ T ) f (θ ) .So, if ω is a fixed number and θ is any angle we have the following periods.2πω2πsec(ωθ) T ωπcot(ωθ) T ω2πω2πcos(ωθ) T ωπtan(ωθ) T ωcsc(ωθ) T sin(ωθ) T 1
Identities and FormulasTangent and Cotangent Identitiestan θ sin θcos θcot θ Half Angle Formulasr1 cos(2θ)sin θ 2r1 cos(2θ)cos θ 2s1 cos(2θ)tan θ 1 cos(2θ)cos θsin θReciprocal Identities1csc θ1cos θ sec θ1tan θ cot θsin θ 1sin θ1sec θ cos θ1cot θ tan θcsc θ Sum and Difference Formulassin(α β) sin α cos β cos α sin βPythagorean Identities2cos(α β) cos α cos β sin α sin β2sin θ cos θ 1tan2 θ 1 sec2 θtan(α β) 1 cot2 θ csc2 θProduct to Sum FormulasEven and Odd Formulassin( θ) sin θcos( θ) cos θtan( θ) tan θ1sin α sin β [cos(α β) cos(α β)]21cos α cos β [cos(α β) cos(α β)]21sin α cos β [sin(α β) sin(α β)]21cos α sin β [sin(α β) sin(α β)]2csc( θ) csc θsec( θ) sec θcot( θ) cot θPeriodic FormulasIf n is an integersin(θ 2πn) sin θcos(θ 2πn) cos θtan(θ πn) tan θcsc(θ 2πn) csc θsec(θ 2πn) sec θcot(θ πn) cot θSum to Product Formulas α βα βsin α sin β 2 sincos22 α βα βsin α sin β 2 cossin22 α βα βcos α cos β 2 coscos22 α βα βcos α cos β 2 sinsin22Double Angle Formulassin(2θ) 2 sin θ cos θcos(2θ) cos2 θ sin2 θ 2 cos2 θ 1 1 2 sin2 θtan(2θ) tan α tan β1 tan α tan β2 tan θ1 tan2 θCofunction Formulas π π Degrees to Radians Formulassin θ cos θcos θ sin θ22If x is an angle in degrees and t is an angle in π π radians then:csc θ sec θsec θ csc θ22 π π πtπx180 t t andx tan θ cotθcot θ tan θ180 x180 π222
Unit Circle(0, 1)( 12 ,( 3)2( 12 , 3)2 22,)22 (60 , π3120 , 2π3 22,)22( 3 1, )2 230 , π6150 , 5π6( 1, 0) 45 , π4135 , 3π43 1, )2 2( 90 , π2 180 , π0 , 2π210 , 7π6( 3, 12 )2( 330 , 11π6225 , 5π4240 , 4π3 2, 22 )2300 , 5π3(( 12 , 3)2270 , 3π2( 12 , (0, 1)F or any ordered pair on the unit circle (x, y) : cos θ x and sin θ yExamplecos ( 7π6)(315 , 7π4 231sin ( 7π6 ) 233)2 3, 21 )2 22, )22(1, 0)
Inverse Trig FunctionsDefinitionInverse PropertiesThese properties hold for x in the domain and θ inthe rangeθ sin 1 (x) is equivalent to x sin θθ cos 1 (x) is equivalent to x cos θθ tan 1 (x) is equivalent to x tan θDomain and RangeFunctionDomainRangeθ sin 1 (x) 1 x 1 1θ cos (x) 1θ tan (x) x sin 1 (sin(θ)) θcos(cos 1 (x)) xcos 1 (cos(θ)) θtan(tan 1 (x)) xtan 1 (tan(θ)) θOther Notationsππ θ 22sin 1 (x) arcsin(x)0 θ π 1 x 1sin(sin 1 (x)) xcos 1 (x) arccos(x)ππ θ 22tan 1 (x) arctan(x)Law of Sines, Cosines, and TangentsβacγαbLaw of SinesLaw of Tangentstan 12 (α β)a b a btan 12 (α β)sin αsin βsin γ abcLaw of Cosinestan 12 (β γ)b c b ctan 12 (β γ)a2 b2 c2 2bc cos αb2 a2 c2 2ac cos βtan 12 (α γ)a c a ctan 21 (α γ)c2 a2 b2 2ab cos γ4
Complex Numbersi a i a, a 0 i2 1 1i3 ii4 1(a bi)(a bi) a2 b2 a2 b2 Complex Modulus(a bi) (c di) a c (b d)i a bi (a bi) (c di) a c (b d)i(a bi) a bi Complex Conjugate(a bi)(c di) ac bd (ad bc)i(a bi)(a bi) a bi 2DeMoivre’s TheoremLet z r(cos θ i sin θ), and let n be a positive integer.Then:z n rn (cos nθ i sin nθ).Example: Let z 1 i, find z 6 .Solution: First write z in polar form.p (1)2 ( 1)2 2 1π 1θ arg(z) tan 14 π π Polar Form: z 2 cos i sin 44r Applying DeMoivre’s Theorem gives : 6 π π 6z 2cos 6 · i sin 6 · 44 3π3π 23 cos i sin 22 8(0 i(1)) 8i5
Finding the nth roots of a number using DeMoivre’s TheoremExample: Find all the complex fourth roots of 4. That is, find all the complex solutions ofx4 4.We are asked to find all complex fourth roots of 4.These are all the solutions (including the complex values) of the equation x4 4.For any positive integer n , a nonzero complex number z has exactly n distinct nth roots.More specifically, if z is written in the trigonometric form r(cos θ i sin θ), the nth roots ofz are given by the following formula. θ 360 kθ 360 k i sin, f or k 0, 1, 2, ., n 1.( ) r cosnnnn1nRemember from the previous example we need to write 4 in trigonometric form by using:pbr (a)2 (b)2.andθ arg(z) tan 1aSo we have the complex number a ib 4 i0.Therefore a 4 and b 0p(4)2 (0)2 4 and0θ arg(z) tan 1 04Finally our trigonometric form is 4 4(cos 0 i sin 0 )So r Using the formula ( ) above with n 4, we can find the fourth roots of 4(cos 0 i sin 0 ) 10360 0360 00 For k 0, 4 4 cos i sin 2 (cos(0 ) i sin(0 )) 24444 10360 10360 14 For k 1, 4 cos i sin 2 (cos(90 ) i sin(90 )) 2i4444 10360 20360 2 For k 2, 4 4 cos i sin 2 (cos(180 ) i sin(180 )) 24444 10360 30360 3 For k 3, 4 4 cos i sin 2 (cos(270 ) i sin(270 )) 2i4444Thus all of the complex roots of x4 4 are: 2, 2i, 2, 2i .6
Formulas for the Conic SectionsCircleStandardF orm : (x h)2 (y k)2 r2W here (h, k) center and r radiusEllipseStandard F orm f or Horizontal M ajor Axis :(x h)2 (y k)2 1a2b2Standard F orm f or V ertical M ajor Axis :(x h)2 (y k)2 1b2a2Where (h, k) center2a length of major axis2b length of minor axis(0 b a)Foci can be found by using c2 a2 b2Where c foci length7
More Conic SectionsHyperbolaStandard F orm f or Horizontal T ransverse Axis :(x h)2 (y k)2 1a2b2Standard F orm f or V ertical T ransverse Axis :(y k)2 (x h)2 1a2b2Where (h, k) centera distance between center and either vertexFoci can be found by using b2 c2 a2Where c is the distance betweencenter and either focus. (b 0)ParabolaVertical axis: y a(x h)2 kHorizontal axis: x a(y k)2 hWhere (h, k) vertexa scaling factor8
f (x)f (x) sin(x)1 32 �37π411π62π4π33π25π37π411π62π 12 22 23 -1 5π 2 Example : sin42 f (x)f (x) cos(x)1 32 2212x0π6π4π3π22π33π45π6π7π65π4 12 22 23 -1 7π 3 Example : cos62 9
π2π2f (x)f (x) tan x 31 33 π3π2π 5π6 4 3 π3 π4 π60 33 1 310xπ6π4π32π33π45π6π
Trigonometric Formula Sheet De nition of the Trig Functions Right Triangle De nition Assume that: 0 ˇ 2 or 0 90 hypotenuse adjacent opposite sin opp hyp csc hyp opp cos adj hyp sec hyp adj tan opp adj cot adj opp Unit Circle De nition Assume can be any angle. x y y x 1 (x;y) sin y 1 csc 1 y cos x 1 sec 1 x tan y x .
