Trigonometric Identities Six Trigonometric Functions Right triangle definitions, where 0 < 𝜃𝜃 < 𝜋𝜋/2 sin 𝜃𝜃 = opp hyp adj hyp opp tan 𝜃𝜃 = adj cos 𝜃𝜃 = Circular function definitions, where 𝜃𝜃 is any angle. 𝑟𝑟 = �𝑥𝑥 2 + 𝑦𝑦 2 𝑦𝑦 𝑟𝑟 𝑥𝑥 cos 𝜃𝜃 = 𝑟𝑟 𝑦𝑦 tan 𝜃𝜃 = 𝑥𝑥 sin 𝜃𝜃 = Negative Angle Identities sin(−𝜃𝜃) = − sin 𝜃𝜃 csc(−𝜃𝜃) = − csc 𝜃𝜃 cos(−𝜃𝜃) = cos 𝜃𝜃 sec(−𝜃𝜃) = sec 𝜃𝜃 Tangent and Cotangent Identities tan 𝜃𝜃 = sin 𝜃𝜃 cos 𝜃𝜃 Pythagorean Identities sin2 𝜃𝜃 + cos2 𝜃𝜃 = 1 cot 2 𝜃𝜃 + 1 = csc 2 𝜃𝜃 cot 𝜃𝜃 = csc 𝜃𝜃 = hyp opp cot 𝜃𝜃 = adj opp sec 𝜃𝜃 = 𝑟𝑟 𝑦𝑦 𝑟𝑟 sec 𝜃𝜃 = 𝑥𝑥 𝑥𝑥 cot 𝜃𝜃 = 𝑦𝑦 csc 𝜃𝜃 = tan(−𝜃𝜃) = − tan 𝜃𝜃 cot(−𝜃𝜃) = − cot 𝜃𝜃 cos 𝜃𝜃 sin 𝜃𝜃 tan2 𝜃𝜃 + 1 = sec 2 𝜃𝜃 Cofunction Identities 𝜋𝜋 sin � − 𝜃𝜃� = cos 𝜃𝜃 2 𝜋𝜋 csc � − 𝜃𝜃� = sec 𝜃𝜃 2 𝜋𝜋 sec � − 𝜃𝜃� = csc 𝜃𝜃 2 hyp adj Reciprocal Identities 1 csc 𝜃𝜃 1 csc 𝜃𝜃 = sin 𝜃𝜃 1 sec 𝜃𝜃 1 sec 𝜃𝜃 = cos 𝜃𝜃 sin 𝜃𝜃 = 1 cot 𝜃𝜃 1 cot 𝜃𝜃 = tan 𝜃𝜃 cos 𝜃𝜃 = tan 𝜃𝜃 = Double Angle Identities sin 2𝜃𝜃 = 2 sin 𝜃𝜃 cos 𝜃𝜃 cos 2𝜃𝜃 = cos2 𝜃𝜃 − sin2 𝜃𝜃 = 2 cos2 𝜃𝜃 − 1 = 1 − 2 sin2 𝜃𝜃 tan 2𝜃𝜃 = 2 tan 𝜃𝜃 1 − tan2 𝜃𝜃 Half Angle Identities 𝜋𝜋 cos � − 𝜃𝜃� = sin 𝜃𝜃 2 𝜋𝜋 tan � − 𝜃𝜃� = cot 𝜃𝜃 2 𝜋𝜋 cot � − 𝜃𝜃� = tan 𝜃𝜃 2 Sum and Difference Formulas 𝐴𝐴 + 𝐵𝐵 𝐴𝐴 − 𝐵𝐵 sin 𝐴𝐴 + sin 𝐵𝐵 = 2 sin � � cos � � 2 2 𝐴𝐴 − 𝐵𝐵 𝐴𝐴 + 𝐵𝐵 � sin � � sin 𝐴𝐴 − sin 𝐵𝐵 = 2 cos � 2 2 𝐴𝐴 − 𝐵𝐵 𝐴𝐴 + 𝐵𝐵 � cos � � cos 𝐴𝐴 + cos 𝐵𝐵 = 2 cos � 2 2 𝐴𝐴 − 𝐵𝐵 𝐴𝐴 + 𝐵𝐵 � sin � � cos 𝐴𝐴 − cos 𝐵𝐵 = −2 sin � 2 2 MVCC Learning Commons IT129 sin tan 𝜃𝜃 1 − cos 𝜃𝜃 = ±� 2 2 𝜃𝜃 1 − cos 𝜃𝜃 = ±� 2 1 + cos 𝜃𝜃 cos 𝜃𝜃 1 + cos 𝜃𝜃 = ±� 2 2 Addition and Subtraction Formulas sin(𝐴𝐴 ± 𝐵𝐵) = sin 𝐴𝐴 cos 𝐵𝐵 ± cos 𝐴𝐴 sin 𝐵𝐵 cos(𝐴𝐴 ± 𝐵𝐵) = cos 𝐴𝐴 cos 𝐵𝐵 ∓ sin 𝐴𝐴 sin 𝐵𝐵 tan(𝐴𝐴 ± 𝐵𝐵) = tan 𝐴𝐴 ± tan 𝐵𝐵 1 ∓ tan 𝐴𝐴 tan 𝐵𝐵 Product Formulas 1 sin 𝐴𝐴 sin 𝐵𝐵 = [cos(𝐴𝐴 − 𝐵𝐵) − cos(𝐴𝐴 + 𝐵𝐵)] 2 1 cos 𝐴𝐴 cos 𝐵𝐵 = [cos(𝐴𝐴 + 𝐵𝐵) + cos(𝐴𝐴 − 𝐵𝐵)] 2 1 sin 𝐴𝐴 cos 𝐵𝐵 = [sin(𝐴𝐴 + 𝐵𝐵) + sin(𝐴𝐴 − 𝐵𝐵)] 2 1 cos 𝐴𝐴 sin 𝐵𝐵 = [sin(𝐴𝐴 + 𝐵𝐵) − sin(𝐴𝐴 − 𝐵𝐵)] 2