© 2005 Paul Dawkins Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2π θ < < or 0 90 θ ° < < ° . opposite sin hypotenuse θ= hypotenuse csc opposite θ= adjacent cos hypotenuse θ= hypotenuse sec adjacent θ= opposite tan adjacent θ= adjacent cot opposite θ= Unit circle definition For this definition θ is any angle. sin 1y y θ= = 1 csc y θ= cos 1x x θ= = 1 sec x θ= tan yx θ= cot xy θ= Facts and Properties Domain The domain is all the values of θ that can be plugged into the function. sin θ , θ can be any angle cos θ, θ can be any angle tan θ, 1 , 0, 1, 2, 2 n n θ π ⎛ ⎞ ≠ + = ± ± ⎜ ⎟ ⎝ ⎠ … csc θ, , 0, 1, 2, n n θ π ≠ = ± ± … sec θ, 1 , 0, 1, 2, 2 n n θ π ⎛ ⎞ ≠ + = ± ± ⎜ ⎟ ⎝ ⎠ … cot θ, , 0, 1, 2, n n θ π ≠ = ± ± … Range The range is all possible values to get out of the function. 1 sin 1 θ − ≤ ≤ csc 1 and csc 1 θ θ ≥ ≤− 1 cos 1 θ − ≤ ≤ sec 1 and sec 1 θ θ ≥ ≤− tan θ −∞ ≤ ≤ ∞ cot θ −∞ ≤ ≤ ∞ Period The period of a function is the number, T, such that ( ) ( ) f T f θ θ + = . So, if ω is a fixed number and θ is any angle we have the following periods. ( ) sin ωθ → 2 T π ω = ( ) cos ω θ → 2 T π ω = ( ) tan ωθ → T πω = ( ) csc ωθ → 2 T π ω = ( ) sec ω θ → 2 T π ω = ( ) cot ωθ → T πω = θ adjacent opposite hypotenuse x y ( ) , x y θ x y 1 © 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities sin cos tan cot cos sin θ θ θ θ θ θ = = Reciprocal Identities 1 1 csc sin sin csc 1 1 sec cos cos sec 1 1 cot tan tan cot θ θ θ θ θ θ θ θ θ θ θ θ = = = = = = Pythagorean Identities 2 2 2 2 2 2 sin cos 1 tan 1 sec 1 cot csc θ θ θ θ 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. ( ) ( ) ( ) ( ) ( ) ( ) sin 2 sin csc 2 csc cos 2 cos sec 2 sec tan tan cot cot n n n n n n θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ + = + = + = + = + = + = Double Angle Formulas ( ) ( ) ( ) 2 2 2 2 2 sin 2 2sin cos cos 2 cos sin 2cos 1 1 2sin 2 tan tan 2 1 tan θ θ θ θ θ θ θ θ θ θ θ == − = − = − = − Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then 180 and 180 180 t x t t x x π π π = ⇒ = = Half Angle Formulas ( ) ( ) ( ) ( ) ( ) ( ) 222 1 sin 1 cos 2 21 cos 1 cos 2 2 1 cos 2 tan 1 cos 2 θ θ θ θ θ 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 α β α β α β α β α β α β α β ± = ± ± = ± ± = ∓ ∓ Product to Sum Formulas ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 sin sin cos cos 21 cos cos cos cos 2 1 sin cos sin sin 21 cos sin sin sin 2 α β α β α β α β α β α β α β α β α β α β α β α β = − − + ⎡ ⎤ ⎣ ⎦ = − + + ⎡ ⎤ ⎣ ⎦ = + + − ⎡ ⎤ ⎣ ⎦ = + − − ⎡ ⎤ ⎣ ⎦ 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 α β α β + − ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + − ⎛ ⎞ ⎛ ⎞ − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + − ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + − ⎛ ⎞ ⎛ ⎞ − =− ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Cofunction Formulas sin cos cos sin 2 2 csc sec sec csc 2 2 tan cot cot tan 2 2 π π θ θ θ θ π π θ θ θ θ π π θ θ θ θ ⎛ ⎞ ⎛ ⎞ − = − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ − = − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ − = − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ © 2005 Paul Dawkins Unit Circle For any ordered pair on the unit circle ( ) , x y : cos x θ= and sin y θ= Example 5 1 5 3 cos sin 3 2 3 2 π π ⎛ ⎞ ⎛ ⎞ = =− ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 3π 4π 6π 2 2 , 2 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 3 1 , 2 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 3 , 2 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 60° 45° 30° 23π 34π 56π 76π 54π 43π 116 π 74π 53π 2π π 32π 0 2 π 1 3 , 2 2 ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ 2 2 , 2 2 ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ 3 1 , 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° 360° 0° x ( ) 1,0 y© 2005 Paul Dawkins Inverse Trig Functions Definition 111 sin is equivalent to sin cos is equivalent to cos tan is equivalent to tan y x x y y x x y y x x y −−− = = = = = = Domain and Range Function Domain Range 1 sin y x− = 1 1 x − ≤ ≤ 2 2 y π π − ≤ ≤ 1 cos y x− = 1 1 x − ≤ ≤ 0 y π ≤ ≤ 1 tan y x− = x −∞ < < ∞ 2 2 y π π − < < Inverse Properties ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 cos cos cos cos sin sin sin sin tan tan tan tan x x x x x x θ θ θ θ θ θ − − − − − − = = = = = = Alternate Notation 111 sin arcsin cos arccos tan 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 2 2 2 2 2 2 2 cos 2 cos 2 cos a b c bc b a c ac c a b ab αβγ = + − = + − = + − Mollweide’s Formula ( ) 12 12 cossin a b c α β γ − + = Law of Tangents ( ) ( ) ( ) ( ) ( ) ( ) 1212 12121212 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 − − = + +− − = + +− − = + + c a b α β γ