Trig Cheat Sheet

Definition of the Trig Functions Right triangle definition For this definition we assume that Unit circle definition π For this definition θ is any angle. 0 <<θ or 0°<θ <°90 . 2 y

( xy, ) 1 hypotenuse y θ opposite x x θ adjacent

opposite hypotenuse y 1 sinθ = cscθ = sinθ ==y cscθ = hypotenuse opposite 1 y adjacent hypotenuse x 1 cosθ = secθ = cosθ ==x secθ = hypotenuse adjacent 1 x opposite adjacent y x tanθ = cotθ = tanθ = cotθ = adjacent opposite x y

Facts and Properties Domain The domain is all the values of θ that Period can be plugged into the function. The period of a function is the number, T, such that f(θθ+=Tf) ( ) . So, if ω sinθ , θ can be any angle is a fixed number and θ is any angle we cosθ , θ can be any angle have the following periods. 1 tanθ , θπ≠nn+,=0,±±1,2,… 2 2π sin (ωθ) → T = cscθ , θπ≠nn,=0,±±1,2,… ω 1 2π secθ , θπ≠nn+,=0,±±1,2,… cos(ωθ) → T = 2 ω π cotθ , θπ≠nn,=0,±±1,2,… tan (ωθ) → T = ω Range 2π csc(ωθ) → T = The range is all possible values to get ω out of the function. 2π sec(ωθ) → T = −1≤≤sin1θ cscθθ≥1andcsc1≤− ω −1≤≤cos1θ secθθ≥1andsec1≤− π cot (ωθ) → T = −∞

© 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas (alternate form) sinθθcos θθ1−cos1 tanθθ==cot sin=±sin2 θθ=−(1cos2( )) cosθθsin 222 Reciprocal Identities θθ1+cos1 11cos=±cos2 θθ=+(1cos2( )) cscθθ==sin 222 sinθθcsc 11θθ1−cos 2 1−cos2( θ ) secθθ==cos tan=±=tan θ cosθθsec 21++cosθθ1cos2( ) 11Sum and Difference Formulas cotθθ==tan tanθθcot sin(α±β) =±sinαcosβcosαβsin Pythagorean Identities cos(α±=β) cosαcosβ∓sinαβsin sin22θθ+=cos1 tanαβ±tan tan22θθ+=1sec tan (αβ±=) 1∓ tanαβtan 1+=cot22θθcsc Product to Sum Formulas Even/Odd Formulas 1 sinαsinβ=cos(α−β) −+cos(αβ) sin(−θ) =−sinθcsc(−θθ) =−csc 2 1 cos(−θ) =cosθsec(−=θθ) sec cosαcosβ=cos(α−β) ++cos(αβ) 2  tan(−θ) =−tanθcot(−θθ) =−cot 1 sinαcosβ=sin(α+β) +−sin (αβ) Periodic Formulas 2 If n is an integer. 1 cosαsinβ=sin(α+β) −−sin (αβ) sin(θ+2πnn) =sinθcsc(θ+=2πθ) csc 2 Sum to Product Formulas cos(θ+2πnn) =cosθsec(θ+=2πθ) sec α+−βαβ tanθ+πnn=tanθcotθ+=πθcot sinαβ+=sin2sincos ( ) ( ) 22 Double Angle Formulas α+−βαβ sinαβ−=sin2cossin  sin(2θ) = 2sinθθcos 22 cos(2θ) =−cos22θθsin α+−βαβ cosαβ+=cos2coscos 2 22 =−2cos1θ  2 α+−βαβ =−12sin θ cosαβ−cos=−2sinsin  22 2tanθ tan2( θ ) = Cofunction Formulas 1−tan2θ ππ Degrees to Radians Formulas sin−θ=cosθcos−=θθsin 22 If x is an angle in degrees and t is an  ππ angle in radians then csc−θ=secθsec−=θθcsc ππtxt180 22 =⇒tx==and ππ 180x180 π tan−θ=cotθcot−=θθtan 22

Unit Circle

y (0,1) π 13 13 , − , 2 22 22 π 22 2π 90° , 22 3 22 − ,  22 3 π  120° 60° 3π 31 4 , 22 31 4  − , 135° 45° π 22 5π 6 6 30° 150°

(−1,0) π 180° 0° 0 (1,0) 360° 2π x

210° 7π 330° 11π 6 225° 6 31 31 −−, 315° ,− 22 5π 22 240° 4 300° 7π 22 4π 270° −−, 5π 4 22 22 3 3π ,− 3 22 2 13 13 −−, ,− 22 22 (0,1− )

For any ordered pair on the unit circle ( xy, ) : cosθ = x and sinθ = y

Example 5ππ153 cos=sin =− 3232

Inverse Trig Functions Definition Inverse Properties y==sin−1 x is equivalent to xysin cos(cos−−11( xx)) ==cos(cos(θθ)) −1 y==cosx is equivalent to xycos sin(sin−−11()xx)==sin(sin ()θθ) y==tan−1 x is equivalent to xytan tantan−−11()xx==tan(tan ()θθ) ( ) Domain and Range Function Domain Range Alternate Notation ππ sin−1 xx= arcsin yx= sin−1 −11≤≤x −≤≤y 22 cos−1 xx= arccos −1 yx= cos −11≤≤x 0 ≤≤y π tan−1 xx= arctan ππ yx= tan −1 −∞

Law of Sines, Cosines and Tangents

c β a

α γ

Law of Sines Law of Tangents sinαsinβγsin ab− tan 1 (αβ− ) == = 2 abc 1 ab++tan 2 (αβ) Law of Cosines 1 bc−tan 2 (βγ−) 222 = a=b+−c2bc cosα 1 bc++tan 2 (βγ) 222 b=a+−c2ac cos β 1 ac−tan 2 (αγ−) 222 = c=a+−b2ab cosγ 1 ac++tan 2 (αγ) Mollweide’s Formula ab+ cos 1 (αβ− ) = 2 1 c sin 2 γ