Trigonometry Cheat Sheet with Unit Circle, Cheat Sheet of Trigonometry

Download Trigonometry Cheat Sheet with Unit Circle and more Cheat Sheet Trigonometry in PDF only on Docsity! Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 π θ< < or 0 90θ° < < ° . oppositesin hypotenuse θ = hypotenusecsc opposite θ = adjacentcos hypotenuse θ = hypotenusesec adjacent θ = oppositetan adjacent θ = adjacentcot opposite θ = Unit circle definition For this definition θ is any angle. sin 1 y yθ = = 1csc y θ = cos 1 x xθ = = 1sec x θ = tan y x θ = cot 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 n nθ π ≠ + = ± ±    … cscθ , , 0, 1, 2,n nθ π≠ = ± ± … secθ , 1 , 0, 1, 2, 2 n nθ π ≠ + = ± ±    … cotθ , , 0, 1, 2,n nθ π≠ = ± ± … Range The range is all possible values to get out of the function. 1 sin 1θ− ≤ ≤ csc 1 and csc 1θ θ≥ ≤ − 1 cos 1θ− ≤ ≤ sec 1 and sec 1θ θ≥ ≤ − tanθ−∞ < < ∞ cotθ−∞ < < ∞ Period The period of a function is the number, T, such that ( ) ( )f T fθ θ+ = . So, if ω is a fixed number and θ is any angle we have the following periods. ( )sin ωθ → 2T π ω = ( )cos ωθ → 2T π ω = ( )tan ωθ → T π ω = ( )csc ωθ → 2T π ω = ( )sec ωθ → 2T π ω = ( )cot ωθ → T π ω = θ adjacent opposite hypotenuse x y ( ),x y θ x y 1 Formulas and Identities Tangent and Cotangent Identities sin costan cot cos sin θ θ θ θ θ θ = = Reciprocal Identities 1 1csc sin sin csc 1 1sec cos cos sec 1 1cot 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 tantan 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 180and 180 180 t x tt x x π π π = ⇒ = = Half Angle Formulas (alternate form) ( )( ) ( )( ) ( ) ( ) 2 2 2 1 cos 1sin sin 1 cos 2 2 2 2 1 cos 1cos cos 1 cos 2 2 2 2 1 cos 21 costan tan 2 1 cos 1 cos 2 θ θ θ θ θ θ θ θ θθ θ θ θ θ − = ± = − + = ± = + −− = ± = + + Sum and Difference Formulas ( ) ( ) ( ) sin sin cos cos sin cos cos cos sin sin tan tantan 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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1sin sin cos cos 2 1cos cos cos cos 2 1sin cos sin sin 2 1cos 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 2 cos sin 2 2 cos cos 2 cos 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    − = − =           − = − =           − = − =       