Trigonometry Formulas and Properties

Reciprocal Identities: Tangent and Cotangent Identities:

sin cos 1 1 tan = cot = sin = csc = cos sin csc sin 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 1𝜃𝜃 1 𝜃𝜃 cos = sec = Pythagorean Identities: sec cos 𝜃𝜃 𝜃𝜃 sin + cos = 1 1 𝜃𝜃 1 𝜃𝜃 tan = cot = tan2 + 1 =2sec cot tan 𝜃𝜃 𝜃𝜃 1 +2cot = csc2 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 2 2 𝜃𝜃 𝜃𝜃

Even/Odd Formulas: Cofunction Formulas: sin( ) = sin cos( ) = cos tan( ) = tan sin = cos csc = sec tan = cot 2 2 2 𝜋𝜋 𝜋𝜋 𝜋𝜋 csc(−𝜃𝜃) = − csc𝜃𝜃 sec(−𝜃𝜃) = sec 𝜃𝜃 cot(−𝜃𝜃) = − cot 𝜃𝜃 cos� − 𝜃𝜃� = sin 𝜃𝜃 sec� − 𝜃𝜃� = csc 𝜃𝜃 cot � − 𝜃𝜃� = tan 𝜃𝜃

2 2 2 𝜋𝜋 𝜋𝜋 𝜋𝜋 −𝜃𝜃 − 𝜃𝜃 −𝜃𝜃 𝜃𝜃 −𝜃𝜃 − 𝜃𝜃 � − 𝜃𝜃� 𝜃𝜃 � − 𝜃𝜃� 𝜃𝜃 � − 𝜃𝜃� 𝜃𝜃 Product to Sum Formulas: Sum to Product Formulas: 1 + sin sin = [cos( ) cos( + )] sin + sin = 2 sin cos 2 2 2 1 𝛼𝛼 +𝛽𝛽 𝛼𝛼 − 𝛽𝛽 cos 𝛼𝛼 cos𝛽𝛽 = [cos(𝛼𝛼 − 𝛽𝛽 )−+ cos(𝛼𝛼 +𝛽𝛽 )] sin 𝛼𝛼 sin 𝛽𝛽 = 2 cos� � sin � � 2 2 2 1 𝛼𝛼 + 𝛽𝛽 𝛼𝛼 − 𝛽𝛽 sin𝛼𝛼 cos𝛽𝛽 = [sin(𝛼𝛼 +− 𝛽𝛽) + sin( 𝛼𝛼 𝛽𝛽)] cos 𝛼𝛼 +−cos𝛽𝛽 = 2 cos� � cos� � 2 2 2 1 𝛼𝛼 +𝛽𝛽 𝛼𝛼 − 𝛽𝛽 cos𝛼𝛼 sin 𝛽𝛽 = [sin(𝛼𝛼 + 𝛽𝛽) sin(𝛼𝛼 − 𝛽𝛽)] 𝛼𝛼 𝛽𝛽 � � � � 2 cos cos = 2 sin sin

2 2 𝛼𝛼 𝛽𝛽 𝛼𝛼 𝛽𝛽 − 𝛼𝛼 − 𝛽𝛽 𝛼𝛼 𝛽𝛽 𝛼𝛼 − 𝛽𝛽 𝛼𝛼 − 𝛽𝛽 − � � � �

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(𝛼𝛼 ± 𝛽𝛽) = cos𝛼𝛼 cos𝛽𝛽 sin𝛽𝛽 sin𝛼𝛼 𝜃𝜃 − 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 � � � cos(2 ) = cos sin tan ± tan 1 + cos 𝛼𝛼tan(𝛽𝛽 ± ) = 𝛼𝛼 𝛽𝛽 ∓ 𝛼𝛼 𝛽𝛽 cos = ± 2 2 1 tan tan 2 2 = 2𝜃𝜃 cos 𝜃𝜃1− 𝜃𝜃 𝛼𝛼 𝛽𝛽 𝜃𝜃 𝜃𝜃 2 𝛼𝛼 𝛽𝛽 � = 1 2 sin ∓ 𝛼𝛼 𝛽𝛽 � � 𝜃𝜃 − Periodic Formulas: 1 cos 2 tan = ± − 𝜃𝜃 sin( + 2 ) = sin csc( + 2 ) = csc 2 1 + cos 2tan 𝜃𝜃 − 𝜃𝜃 tan 2 = � � � 1 tan

cos(𝜃𝜃 + 2𝜋𝜋𝜋𝜋) = cos𝜃𝜃 sec(𝜃𝜃 + 2𝜋𝜋𝜋𝜋) = sec𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 2 − 𝜃𝜃 tan(𝜃𝜃 + 𝜋𝜋𝜋𝜋) = tan 𝜃𝜃 cot(𝜃𝜃 + 𝜋𝜋𝜋𝜋) = cot 𝜃𝜃 𝜃𝜃 𝜋𝜋𝜋𝜋 𝜃𝜃 𝜃𝜃 𝜋𝜋𝜋𝜋 𝜃𝜃

Updated: October 2019

Trigonometric Functions:

Right Triangle: Unit Circle: y

hypotenuse ( , )

opposite 𝑥𝑥 𝑦𝑦 r x θ

adjacent opposite hypotnuse sin = csc = hypotnuse opposite sin = csc = θ θ 𝑦𝑦 𝑟𝑟 adjacent hypotnuse 𝜃𝜃 𝜃𝜃 cos = sec = hypotnuse adjacent cos = 𝑟𝑟 sec = 𝑦𝑦 θ θ 𝑥𝑥 𝑟𝑟 opposite adjacent 𝜃𝜃 𝜃𝜃 tan = cot = 𝑟𝑟 𝑥𝑥 adjacent opposite tan = cot = θ θ 𝑦𝑦 𝑥𝑥 𝜃𝜃 𝜃𝜃 𝑥𝑥 𝑦𝑦

Inverse Trigonometric Functions:

Definition: Alternative Definition: = sin = sin sin =arcsin −1 −1 𝑦𝑦 = cos 𝑥𝑥 �same����as 𝑥𝑥 = cos𝑦𝑦 cos 𝑥𝑥 =arccos 𝑥𝑥

−1 −1 𝑦𝑦 𝑥𝑥 �same����as 𝑥𝑥 𝑦𝑦 tan 𝑥𝑥 =arctan 𝑥𝑥 = tan = tan Law of Sines: = = −1 sin 𝛼𝛼 sin 𝛽𝛽 sin 𝛾𝛾 −1 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑥 �𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠���𝑎𝑎�𝑎𝑎 𝑥𝑥 𝑦𝑦 𝑎𝑎 𝑏𝑏 𝑐𝑐 Domain and Range: Law of Cosines: = + 2 cos Function Domain Range 2 2 2 = sin 1 1 𝑎𝑎 𝑏𝑏 𝑐𝑐 − 𝑏𝑏𝑏𝑏 𝛼𝛼 2 2 = + 2 cos −1 𝜋𝜋 𝜋𝜋 𝑦𝑦 𝑥𝑥 − ≤ 𝑥𝑥 ≤ − ≤ 𝑦𝑦 ≤ 2 2 2 = cos 1 1 0 𝑏𝑏 = 𝑎𝑎 + 𝑐𝑐 −2 𝑎𝑎𝑎𝑎cos 𝛽𝛽 −1 𝑦𝑦 𝑥𝑥 − ≤ 𝑥𝑥 ≤ ≤ 𝑦𝑦 ≤ 𝜋𝜋 = tan < < 2 2 2 2 𝑐𝑐 𝑎𝑎 𝑏𝑏 − 𝑎𝑎𝑎𝑎 𝛾𝛾 −1 𝜋𝜋 𝜋𝜋 𝑦𝑦 𝑥𝑥 −∞ ≤ 𝑥𝑥 ≤ ∞ − 𝑦𝑦 Law of Tangents: = cot 0 1 tan ( ) −1 = 2 𝑦𝑦 𝑥𝑥 −∞ ≤ 𝑥𝑥 ≤ ∞ ≤ 𝑦𝑦 ≤ 𝜋𝜋 + 1 = sec 1, 1 0 < , < tan (𝛼𝛼+ −𝛽𝛽) 2 2 𝑎𝑎 −𝑏𝑏 2 −1 𝜋𝜋 𝜋𝜋 𝑦𝑦 𝑥𝑥 𝑥𝑥 ≤ − 𝑥𝑥 ≥ ≤ 𝑦𝑦 𝑦𝑦 ≤ 𝜋𝜋 𝑎𝑎 𝑏𝑏 1 = csc 1, 1 < 0 , 0 < tan (𝛼𝛼 𝛽𝛽) 2 2 = 2 −1 1 𝜋𝜋 𝜋𝜋 + tan ( + ) 𝑦𝑦 𝑥𝑥 𝑥𝑥 ≤ − 𝑥𝑥 ≥ − ≤ 𝑦𝑦 𝑦𝑦 ≤ 𝑏𝑏 −𝑐𝑐 2 𝛽𝛽 −𝛾𝛾 𝑏𝑏 𝑐𝑐 1 tan (𝛽𝛽 𝛾𝛾) Inverse Properties: = 2 1 + tan ( + ) 𝑎𝑎 −𝑐𝑐 2 𝛼𝛼 −𝛾𝛾 sin (sin ( )) = sin (sin( )) = 𝑎𝑎 𝑐𝑐 −1 −1 𝛼𝛼 𝛾𝛾 cos (cos (𝑥𝑥 )) =𝑥𝑥 cos (cos(𝜃𝜃 )) =𝜃𝜃 −1 −1 tan (tan (𝑥𝑥)) = 𝑥𝑥 tan (tan(𝜃𝜃)) = 𝜃𝜃 −1 −1 𝑥𝑥 𝑥𝑥 𝜃𝜃 𝜃𝜃 Updated: October 2019