Trigonometry Formulas and Properties Reciprocal Identities: Tangent and Cotangent Identities: sin cos 1 1 tan = cot = sin = csc = cos sin csc sin 1 1 cos = sec = sec cos Pythagorean Identities: sin + cos = 1 1 1 tan = cot = tan2 + 1 =2sec cot tan 2 2 1 + cot = csc 2 2 Even/Odd Formulas: Cofunction Formulas: sin( ) = sin cos( ) = cos tan( ) = tan sin = cos csc = sec tan = cot 2 2 2 csc(−) = − csc sec(−) = sec cot(−) = − cot cos� − � = sin sec� − � = csc cot � − � = tan 2 2 2 − − − − − � − � � − � � − � Product to Sum Formulas: Sum to Product Formulas: 1 + sin sin = [cos( ) cos( + )] sin + sin = 2 sin cos 2 2 2 1 + − cos cos = [cos( − )−+ cos( + )] sin sin = 2 cos� � sin � � 2 2 2 1 + − sin cos = [sin( +− ) + sin( )] cos +−cos = 2 cos� � cos� � 2 2 2 1 + − cos sin = [sin( + ) sin( − )] � � � � 2 cos cos = 2 sin sin 2 2 − − − − − � � � � Sum and Difference Formulas: Half-Angle Formulas: Double Angle Formulas: ( ) sin( ± ) = sin cos ± sin cos 1 cos sin 2 = 2 sin cos sin = ± 2 2 cos( ± ) = cos cos sin sin − � � � cos(2 ) = cos sin tan ± tan 1 + cos tan( ± ) = ∓ cos = ± 2 2 1 tan tan 2 2 = 2 cos 1− 2 � = 1 2 sin ∓ � � − Periodic Formulas: 1 cos 2 tan = ± − sin( + 2 ) = sin csc( + 2 ) = csc 2 1 + cos 2tan − tan 2 = � � � 1 tan cos( + 2) = cos sec( + 2) = sec 2 − tan( + ) = tan cot( + ) = cot Updated: October 2019 Trigonometric Functions: Right Triangle: Unit Circle: y hypotenuse ( , ) opposite r x θ adjacent opposite hypotnuse sin = csc = hypotnuse opposite sin = csc = θ θ adjacent hypotnuse cos = sec = hypotnuse adjacent cos = sec = θ θ opposite adjacent tan = cot = adjacent opposite tan = cot = θ θ Inverse Trigonometric Functions: Definition: Alternative Definition: = sin = sin sin =arcsin −1 −1 = cos �same����as = cos cos =arccos −1 −1 �same����as tan =arctan = tan = tan Law of Sines: = = −1 sin sin sin −1 ������ Domain and Range: Law of Cosines: = + 2 cos Function Domain Range 2 2 2 = sin 1 1 − 2 2 = + 2 cos −1 − ≤ ≤ − ≤ ≤ 2 2 2 = cos 1 1 0 = + −2 cos −1 − ≤ ≤ ≤ ≤ = tan < < 2 2 2 2 − −1 −∞ ≤ ≤ ∞ − Law of Tangents: = cot 0 1 tan ( ) −1 = 2 −∞ ≤ ≤ ∞ ≤ ≤ + 1 = sec 1, 1 0 < , < tan (+ −) 2 2 − 2 −1 ≤ − ≥ ≤ ≤ 1 = csc 1, 1 < 0 , 0 < tan ( ) 2 2 = 2 −1 1 + tan ( + ) ≤ − ≥ − ≤ ≤ − 2 − 1 tan ( ) Inverse Properties: = 2 1 + tan ( + ) − 2 − sin (sin ( )) = sin (sin( )) = −1 −1 cos (cos ( )) = cos (cos( )) = −1 −1 tan (tan ()) = tan (tan()) = −1 −1 Updated: October 2019 .