Trig Cheat Sheet

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Trig Cheat Sheet Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas (alternate form) sinθθcos θθ1−cos1 Definition of the Trig Functions tanθθ==cot sin=±sin2 θθ=−(1cos2( )) Right triangle definition cosθθsin 222 Reciprocal Identities For this definition we assume that Unit circle definition θθ1+cos1 π For this definition θ is any angle. 11cos=±cos2 θθ=+(1cos2( )) 0 <<θ or 0°<θ <°90 . cscθθ==sin 222 2 y sinθθcsc 1−cos2θ 11θθ1−cos 2 ( ) secθθ==cos tan=±=tan θ ( xy, ) cosθθsec 21++cosθθ1cos2( ) 1 11Sum and Difference Formulas hypotenuse y θ cotθθ==tan sinα±β=±sinαcosβcosαβsin opposite x tanθθcot ( ) x Pythagorean Identities cos(α±=β) cosαcosβ∓sinαβsin θ sin22θθ+=cos1 tanαβ±tan adjacent 22 tan (αβ±=) tanθθ+=1sec 1tanαβtan 22 ∓ opposite hypotenuse y 1 1+=cotθθcsc Product to Sum Formulas sinθ = cscθ = sinθ ==y cscθ = hypotenuse opposite 1 y Even/Odd Formulas 1 sinαsinβ=cos(α−β) −+cos(αβ) adjacent hypotenuse x 1 sin(−θ) =−sinθcsc(−θθ) =−csc 2 cosθ = secθ = cosθ ==x secθ = hypotenuse adjacent 1 x 1 cos(−θ) =cosθsec(−=θθ) sec cosαcosβ=cos(α−β) ++cos(αβ) opposite adjacent y x 2 tanθ = cotθ = tanθ = cotθ = tan(−θ) =−tanθcot(−θθ) =−cot adjacent opposite x y 1 sinαcosβ=sin(α+β) +−sin(αβ) Periodic Formulas 2 Facts and Properties If n is an integer. 1 cosαsinβ=sin(α+β) −−sin(αβ) Domain sin(θ+2πnn) =sinθcsc(θ+=2πθ) csc 2 The domain is all the values of θ that Period Sum to Product Formulas cos(θ+2πnn) =cosθsec(θ+=2πθ) sec can be plugged into the function. The period of a function is the number, α+−βαβ tanθ+πnn=tanθcotθ+=πθcot sinαβ+=sin2sincos T, such that f(θθ+=Tf) ( ) . So, if ω ( ) ( ) 22 sinθ , θ can be any angle Double Angle Formulas is a fixed number and θ is any angle we α+−βαβ cosθ , θ can be any angle have the following periods. sinαβ−=sin2cossin  sin(2θ) =2sinθθcos 22 1 tanθ , θπ≠nn+,=0,±±1,2,… 22 2 2π cos(2θ) =−cosθθsin α+−βαβ sin(ωθ) → T = cosαβ+=cos2coscos 2 22 cscθ , θπ≠nn,=0,±±1,2,… ω =−2cos1θ  1 2π 2 α+−βαβ secθ , θπ≠nn+,=0,±±1,2,… cos (ωθ) → T = =−12sin θ cosαβ−cos=−2sinsin 2 ω 22 2tanθ θπ≠nn,=0,±±1,2, π tan2θ= Cofunction Formulas cotθ , … tan (ωθ) → T = ( ) 2 ω 1−tan θ ππ Range 2π Degrees to Radians Formulas sin−θ=cosθcos−=θθsin csc(ωθ) → T = 22 The range is all possible values to get ω If x is an angle in degrees and t is an ππ out of the function. 2π angle in radians then csc−θ=secθsec−=θθcsc sec(ωθ) → T = 22 −1≤≤sin1θ cscθθ≥1andcsc1≤− ω ππtxt180 =⇒tx==and ππ −1≤≤cos1θ secθθ≥1andsec1≤− π 180x180 π tan−θ=cotθcot−=θθtan cot (ωθ) → T = 22 −∞

Unit Circle Inverse Trig Functions Definition Inverse Properties y −1 −−11 (0,1) y==sinx is equivalent to xysin cos(cos( xx)) ==cos(cos(θθ)) −1 π 13 −−11 , y==cosx is equivalent to xycos sinsinxx==sinsin θθ 13 22 ( ()) ( ()) −, 2  −1 22 y==tanx is equivalent to xytan −−11 π 22 tan(tan()xx)==tan(tan ()θθ) 2π 90° , 22 3 22 − ,  22 3 π Domain and Range  120° 60° Alternate Notation 3π 31 Function Domain Range 4 , −1 22 ππ sinxx= arcsin 31 4  yx=sin−1 −11≤≤x −≤≤y − , 135° 45° π −1 22 5π 22 cosxx= arccos 6 −1 6 30° yx=cos −11≤≤x 0 ≤≤y π tan−1 xx= arctan 150° ππ yx=tan −1 −∞

(−1,0) π 180° 0° 0 (1,0) Law of Sines, Cosines and Tangents 360° 2π x

c β a 210° 7π 330° 11π 6 225° 6 31 31 α γ −−, 315° ,− 22 5π 22 240° 300° 7π 4 270° 22 4π b −−, 5π 4 22 22 ,− 3 3π 22 3  Law of Sines Law of Tangents 13 2 −−, 13 1  ,− sinαsinβγsin ab− tan 2 (αβ−) 22 22 == = abc 1 (0,1− ) ab++tan 2 (αβ) Law of Cosines 1 bc−tan 2 (βγ−) 222 = a=b+−c2bc cosα 1 bc++tan 2 (βγ) 222 For any ordered pair on the unit circle ( xy, ) : cosθ = x and sinθ = y b=a+−c2ac cos β 1 ac−tan 2 (αγ−) 222 = c=a+−b2ab cosγ ac++tan 1 (αγ) Example 2 Mollweide’s Formula 5ππ153 1 cos=sin=− ab+ cos 2 (αβ−) 3232 = c sin 1 γ 2