Trigo Sheet Cheat :D

Trigo Sheet Cheat :D

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  θ = y= y csc 1 θ = x = x sec 1 θ = cot x © 2005 Paul Dawkins Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that π 0 <θ < or 0° <θ < 90° . 2 sin opposite θ = csc hypotenuse hypotenuse opposite θ = cos adjacent θ = sec hypotenuse hypotenuse adjacent θ = tan opposite θ = cot adjacent adjacent opposite θ = Unit circle definition For this definition θ is any angle. sin 1 y θ = cos 1 x θ = tan y x y θ = Facts and Properties Domain The domain is all the values of θ that can be plugged into the function. sinθ , θ can be any angle cosθ , θ can be any angle tanθ , 1 , 0, 1, 2, 2 θ ≠ ⎛⎜ n + ⎞⎟π n = ± ± ⎝ ⎠ … cscθ , θ ≠ nπ , n = 0,±1, ± 2,… secθ , 1 , 0, 1, 2, 2 θ ≠ ⎛⎜ n + ⎞⎟π n = ± ± ⎝ ⎠ … cotθ , θ ≠ nπ , n = 0,±1, ± 2,… Range The range is all possible values to get out of the function. −1≤ sinθ ≤1 cscθ ≥1 and cscθ ≤ −1 −1≤ cosθ ≤1 secθ ≥1 and secθ ≤ −1 −∞ ≤ tanθ ≤ ∞ −∞ ≤ cotθ ≤ ∞ Period The 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. sin (ωθ ) → T 2π ω = cos (ωθ ) → T 2 π ω = tan (ωθ ) → T π ω = csc(ωθ ) → T 2π ω = sec(ωθ ) → T 2 π ω = cot (ωθ ) → T π ω = θ adjacent opposite hypotenuse x y (x, y) θ x y 1 θ θ θ θ ± α = β α ± β α β α ± = β α β β α α β sin sin 1 cos cos α β α β α β = ⎡⎣ − − + ⎤⎦ = ⎡⎣ − + + ⎤⎦ = ⎣⎡ + + − ⎦⎤ = ⎡⎣ + − − ⎤⎦ α β α β α β α β α β α β α β α β α β ⎛ + ⎞ ⎛ − ⎞ + = ⎜ ⎟ ⎜ ⎟ α β α β ⎝ ⎠ ⎝ ⎠ ⎛ + ⎞ ⎛ − ⎞ − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ + ⎞ ⎛ − ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ + ⎞ ⎛ − ⎞ − =− ⎜ ⎟ ⎜ ⎟ π π ⎛ − ⎞ = ⎛ − ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ − ⎞ = ⎛ − ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ − ⎞ = ⎛ − ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ θ θ θ θ π π θ θ θ θ π π θ θ θ θ © 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities tan sin cot cos sin 1 1 cos 2 θ θ = θ = θ cos sin θ θ Reciprocal Identities csc 1 sin 1 = θ = θ sin csc θ θ sec = 1 cos = 1 θ θ cos sec θ θ cot = 1 tan = 1 θ θ tan cot θ θ Pythagorean Identities 2 2 sin cos 1 tan 1 sec 1 cot csc + θ = θ + = θ θ 2 2 2 2 + = θ θ Even/Odd Formulas ( ) ( ) ( ) ( ) ( ) ( ) sin − = − sin csc − = − csc cos − = cos sec − = sec tan − = − tan cot − = − cot θ θ θ θ θ θ θ θ θ θ θ θ Periodic Formulas If n is an integer. ( n ) ( n ) ( n ) ( n ) ( n ) ( n ) sin 2 sin csc 2 csc cos 2 cos sec 2 sec tan tan cot cot θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ + = + = + = + = + = + = Double Angle Formulas sin ( 2 ) 2sin cos cos ( 2 ) cos sin θ θ θ θ θ θ ( ) = = − = − = − = 2 2 2 2cos 1 1 2sin θ 2 θ θ 2 tan 2 2 tan 1 tan θ θ − Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then t t x and x 180 t x π π 180 180 π = ⇒ = = Half Angle Formulas ( ( )) ( ( )) ( ) ( ) 2 2 2 2 cos 1 1 cos 2 2 1 cos 2 tan θ 1 cos 2 θ θ = − = + − = + Sum and Difference Formulas ( ) ( ) ( ) sin sin cos cos sin cos cos cos sin sin tan tan ± tan 1 tan tan α β α β ± = ∓ ∓ Product to Sum Formulas ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 cos cos 1 cos cos 2 sin cos 1 sin sin 2 cos sin 1 sin sin 2 Sum to Product Formulas sin sin 2sin cos 2 2 sin sin 2cos sin 2 2 cos cos 2cos cos 2 2 cos cos 2sin sin 2 2 α β α β α β α β α β α β α β α β α β α β ⎝ ⎠ ⎝ ⎠ Cofunction Formulas sin cos cos sin 2 2 csc sec sec csc 2 2 tan cot cot tan 2 2
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  π ⎛ ⎞ ⎜⎜ ⎟⎟ ⎝ ⎠ π ⎛ 2 , 2 ⎞ ⎜⎜ ⎝ 2 2 ⎟⎟ ⎠ π π 0° 360° ⎛ ⎞ ⎜ − ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ − ⎟ ⎝ ⎠ ⎛ π ⎞ ⎛ π ⎞ ⎜ ⎟ = ⎜ ⎟ = − ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎜⎜ ⎟⎟ ⎝ ⎠ (1,0) ⎛ ⎞ ⎜ − ⎟ ⎝ ⎠ © 2005 Paul Dawkins Unit Circle ⎛ ⎞ ⎜− ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ − ⎟ ⎝ ⎠ 3,1 2 2 ⎛ ⎞ ⎜− − ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ − − ⎟ ⎝ ⎠ ⎛ ⎞ ⎜− − ⎟ ⎝ ⎠ For any ordered pair on the unit circle ( x, y) : cosθ = x and sinθ = y Example cos 5 1 sin 5 3 3 2 3 2 3 4 6 3,1 2 2 1, 3 2 2 60° 45° 30° 2 3 π 3 4 π 5 6 π 7 6 π 5 4 π 4 3 π 11 6 π 7 4 π 5 3 π 2 π 3 2 π 0 2π 1, 3 2 2 2 , 2 2 2 ⎛ ⎞ ⎜ − ⎟ ⎝ ⎠ 3, 1 2 2 2 , 2 2 2 1, 3 2 2 3, 1 2 2 2 , 2 2 2 1, 3 2 2 (0,1) (0,−1) (−1,0) 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° x y x x x x x x cos cos cos cos sin sin sin sin tan tan tan tan = = = = = = © 2005 Paul Dawkins Inverse Trig Functions Definition 1 y sin x is equivalent to x sin y y cos x is equivalent to x cos y y tan x is equivalent to x tan y = = = 1 = = 1 = Domain and Range Function Domain Range y = sin−1 x −1≤ x ≤1 − − − y π π − ≤ ≤ 2 2 y = cos−1 x −1≤ x ≤1 0 ≤ y ≤π y = tan−1 x −∞ < x < ∞ π π − < < y 2 2 Inverse Properties ( 1 ( )) 1 ( ( )) ( 1 ( )) 1 ( ( )) ( 1 ( )) 1 ( ( )) θ θ θ θ θ θ − − − − − − Alternate Notation 1 sin arcsin cos 1 arccos tan 1 arctan x x x x x x − − − = = = Law of Sines, Cosines and Tangents Law of Sines sin sin sin a b c α β γ = = Law of Cosines 2 2 2 a b c bc b a c ac c a b ab = + − 2 = 2 + 2 − 2 = 2 + 2 − Mollweide’s Formula 2 cos 2 cos 2 cos α β γ 1 ( ) 2 + − 1 2 cos sin a b c α β γ = c a Law of Tangents ( ) ( ) ( ) ( ) ( ) ( ) − − 1 2 1 2 1 2 1 2 1 2 1 2 tan tan tan tan tan tan a b a b b c b c a c a c α β α β β γ β γ α γ α γ = + + − − = + + − − = + + b α β γ