Formulas /திரிகோணகணித சமன்பாடுகள்
♣ $sin^{ 2 }\alpha \quad +\quad cos^{ 2 }\alpha \quad =\quad 1$
♣ $sin^{ 2 }\alpha \quad +\quad cos^{ 2 }\alpha \quad =\quad 1$
♣ $1\quad +\quad tan^{ 2 }\alpha \quad =\quad sec^{ 2 }\alpha \\ $
♣ $1\quad +\quad cot^{ 2 }\alpha \quad =\quad cosec^{ 2 }\alpha \\ $
☀$tan\alpha =\frac { sin\alpha }{ cos\alpha } $
☀$cot\alpha =\frac { cos\alpha }{ sin\alpha } $
Addition and Subtraction Formulas
♣ $sin(\alpha +\beta )\quad =\quad sin\alpha .cos\beta \quad +\quad sin\beta .cos\alpha $
♣ $sin(\alpha -\beta )\quad =\quad sin\alpha .cos\beta \quad -\quad sin\beta .cos\alpha $
♣ $cos(\alpha +\beta )\quad =\quad cos\alpha .cos\beta \quad -\quad sin\beta .sin\alpha $
♣ $cos(\alpha -\beta )\quad =\quad cos\alpha .cos\beta \quad +\quad sin\beta .sin\alpha $
♣ $tan(\alpha +\beta )\quad =\quad \frac { tan\alpha \quad +\quad tan\quad \beta }{ 1-tan\alpha .tan\beta } $
♣ $tan(\alpha -\beta )\quad =\quad \frac { tan\alpha \quad -\quad tan\quad \beta }{ 1+tan\alpha .tan\beta } $
Double Angle Formulas
♣ $sin2\alpha =sin\alpha .cos\alpha $
♣ $cos2\alpha \quad =\quad cos^{ 2 }\alpha \quad +\quad sin^{ 2 }\alpha \quad =\quad 1\quad -\quad 2sin^{ 2 }\alpha \quad =\quad 2cos^{ 2 }\alpha \quad -\quad 1$
♣ $tan2\alpha \quad =\quad \frac { 2tan\alpha }{ 1-tan^{ 2 }\alpha } \quad =\quad \frac { 2 }{ cot\alpha \quad -\quad sin\alpha } $
Triple angle Formulas
♣ $sin3\alpha =3sin\alpha -4sin^{ 3 }\alpha $
♣ $cos3\alpha =4cos^{ 3 }\alpha \quad -\quad 3cos\alpha \\ \\ $
♣ $tan3\alpha \quad =\frac { 3tan\alpha -tan^{ 3 }\alpha }{ 1-3tan^{ 2 }\alpha } \\ \\ $
Half Angle Tangent Identities
♣ $sin\alpha \quad =\quad \frac { 2tan\frac { \alpha }{ 2 } }{ 1+tan^{ 2 }\frac { \alpha }{ 2 } } $
♣ $cos\alpha \quad =\quad \frac { 1-tan^{ 2 }\frac { \alpha }{ 2 } }{ 1+tan^{ 2 }\frac { \alpha }{ 2 } } $
♣ $tan\alpha \quad =\quad \frac { 2tan\frac { \alpha }{ 2 } }{ 1-tan^{ 2 }\frac { \alpha }{ 2 } } $
Transforming of Trigonometric Expressions to Product
♣ $sin\alpha \quad +\quad \quad sin\beta \quad =2sin\frac { \alpha +\beta }{ 2 } cos\frac { \alpha -\beta }{ 2 } $
♣ $sin\alpha \quad -\quad \quad sin\beta \quad =2cos\frac { \alpha +\beta }{ 2 } sin\frac { \alpha -\beta }{ 2 } $
♣ $cos\alpha \quad +\quad \quad cos\beta \quad =2sin\frac { \alpha +\beta }{ 2 } cos\frac { \alpha -\beta }{ 2 } $
♣ $cos\alpha \quad +\quad \quad cos\beta \quad =-2sin\frac { \alpha +\beta }{ 2 } sin\frac { \alpha -\beta }{ 2 } =2sin\frac { \alpha +\beta }{ 2 } sin\frac { \beta -\alpha }{ 2 } $
Addition and Subtraction Formulas
♣ $sin(\alpha +\beta )\quad =\quad sin\alpha .cos\beta \quad +\quad sin\beta .cos\alpha $
♣ $sin(\alpha -\beta )\quad =\quad sin\alpha .cos\beta \quad -\quad sin\beta .cos\alpha $
♣ $cos(\alpha +\beta )\quad =\quad cos\alpha .cos\beta \quad -\quad sin\beta .sin\alpha $
♣ $cos(\alpha -\beta )\quad =\quad cos\alpha .cos\beta \quad +\quad sin\beta .sin\alpha $
♣ $tan(\alpha +\beta )\quad =\quad \frac { tan\alpha \quad +\quad tan\quad \beta }{ 1-tan\alpha .tan\beta } $
♣ $tan(\alpha -\beta )\quad =\quad \frac { tan\alpha \quad -\quad tan\quad \beta }{ 1+tan\alpha .tan\beta } $
♣ $sin2\alpha =sin\alpha .cos\alpha $
♣ $cos2\alpha \quad =\quad cos^{ 2 }\alpha \quad +\quad sin^{ 2 }\alpha \quad =\quad 1\quad -\quad 2sin^{ 2 }\alpha \quad =\quad 2cos^{ 2 }\alpha \quad -\quad 1$
♣ $tan2\alpha \quad =\quad \frac { 2tan\alpha }{ 1-tan^{ 2 }\alpha } \quad =\quad \frac { 2 }{ cot\alpha \quad -\quad sin\alpha } $
Triple angle Formulas
♣ $sin3\alpha =3sin\alpha -4sin^{ 3 }\alpha $
♣ $cos3\alpha =4cos^{ 3 }\alpha \quad -\quad 3cos\alpha \\ \\ $
♣ $tan3\alpha \quad =\frac { 3tan\alpha -tan^{ 3 }\alpha }{ 1-3tan^{ 2 }\alpha } \\ \\ $
Half Angle Tangent Identities
♣ $sin\alpha \quad =\quad \frac { 2tan\frac { \alpha }{ 2 } }{ 1+tan^{ 2 }\frac { \alpha }{ 2 } } $
♣ $cos\alpha \quad =\quad \frac { 1-tan^{ 2 }\frac { \alpha }{ 2 } }{ 1+tan^{ 2 }\frac { \alpha }{ 2 } } $
♣ $tan\alpha \quad =\quad \frac { 2tan\frac { \alpha }{ 2 } }{ 1-tan^{ 2 }\frac { \alpha }{ 2 } } $
Transforming of Trigonometric Expressions to Product
♣ $sin\alpha \quad +\quad \quad sin\beta \quad =2sin\frac { \alpha +\beta }{ 2 } cos\frac { \alpha -\beta }{ 2 } $
♣ $sin\alpha \quad -\quad \quad sin\beta \quad =2cos\frac { \alpha +\beta }{ 2 } sin\frac { \alpha -\beta }{ 2 } $
♣ $cos\alpha \quad +\quad \quad cos\beta \quad =2sin\frac { \alpha +\beta }{ 2 } cos\frac { \alpha -\beta }{ 2 } $
♣ $cos\alpha \quad +\quad \quad cos\beta \quad =-2sin\frac { \alpha +\beta }{ 2 } sin\frac { \alpha -\beta }{ 2 } =2sin\frac { \alpha +\beta }{ 2 } sin\frac { \beta -\alpha }{ 2 } $
Comments
Post a Comment