CI VIL AN D D DE PA RT ME NT ENGR. MARJON CASIL GONTIÑAS, CE LI E IDENTITIES AL TRIGONOMETRIC TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS CONCEPT AND CONVERSION OF RADIANS AND DEGREES TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS CONCEPT AND CONVERSION OF RADIANS AND DEGREES TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS TRIGONOMETRIC RATIOS sin � cos � tan � csc � sec � cot � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS CIVIL AND ALLIED DEPARTMENT CONCEPT AND CONVERSION OF RADIANS AND DEGREES “THE UNIT CIRCLE” TRIGONOMETRIC IDENTITIES VALUES OF TRIGONOMETRIC FUNCTIONS FOR CERTAIN ANGLES ENGR. MARJON CASIL GONTIÑAS Angle � in Degrees 0° 30° 45° 60° 90° 180° 270° 360° ��� � ��� � ��� � 1 2 2 2 3 2 3 3 1 3 2 2 2 1 2 0 1 0 ��� � ��� � ��� � 3 2 3 3 2 - 1 - 1 1 - 3 3 2 0 3 0 - 2 3 3 0 -1 0 - -1 - -1 0 - 0 - -1 0 1 0 - 1 - 2 2 1 Angle � in Radians 0 1 � 6 1 � 4 1 � 3 1 � 2 � 3 � 2 2� ENGR. MARJON CASIL GONTIÑAS TRIGONOMETRIC IDENTITIES FUNDAMENTAL IDENTITIES � ��� � = ��� � � ��� � = ��� � � ��� � ��� � = = ��� � ��� � ��� � ��� � = ��� � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS PYTHAGOREAN TRIGONOMETRIC IDENTITIES ���� � + ���� � = � Divide both sides by sin2 �, then sin2 � + cos2 � 1 = 2 sin � sin2 � � + ���� � = ���� � Divide both sides by cos2 �, then sin2 � + cos2 � 1 = 2 cos � cos2 � ���� � + � = ���� � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS REDUCTION RELATIONS Angle Sine Cosine Tangent Co-tangent Secant Co-secant −� − sin � cos � − tan � − cot � sec � − csc � 90° + � cos � − sin � − cot � − tan � − csc � sec � tan � cot � 90° − � 180° − � cos � sin � cot � sin � − cos � − tan � 270° − � − cos � − sin � cot � 360° − � − sin � 180° + � 270° + � − sin � − cos � − cos � sin � cos � − cot � − tan � tan � csc � sec � − cot � − sec � tan � − csc � − sec � sec � − csc � − tan � − cot � − sec � csc � csc � − csc � − sec � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS SUM AND DIFFERENCE OF TWO ANGLES ��� � ± � = ��� ���� � ± ��� ���� � ��� � ± � = ��� ���� � ∓ ��� ���� � ��� � ± ��� � ��� � ± � = � ∓ ��� ���� � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS DOUBLE-ANGLE IDENTITIES ��� �� = ���� ���� � ��� �� = ���� � − ���� � ��� �� = ����� � − � ��� �� = � − ����� � ���� � ��� �� = � − ���� � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS HALF-ANGLE IDENTITIES � − ��� � � ��� =± � � � + ��� � � ��� =± � � � − ��� � � − ��� � � ��� � ��� =± = = � + ��� � � ��� � � + ��� � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS SUM AND DIFFERENCE IDENTITIES � � ��� � + ��� � = ���� � + � ��� � − � � � � � ��� � − ��� � = ���� � + � ��� � − � � � � � ��� � + ��� � = ���� � + � ��� � − � � � � � ��� � − ��� � =− ���� � + � ��� � − � � � TRIGONOMETRIC IDENTITIES ENGR. MARJON CASIL GONTIÑAS PRODUCT IDENTITIES � � ��� ���� � = ��� � − � − ��� � + � � � � � ��� ���� � = ��� � − � + ��� � + � � � � � ��� ���� � = ��� � − � + ��� � + � � �