Trigonometrical Basic Formulas

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles

$\displaystyle (1)\quad { Sin }^{ 2 }A+{ Cos }^{ 2 }A=1\\ \Rightarrow { Cos }^{ 2 }A=1-{ Sin }^{ 2 }A\qquad or\quad { Sin }^{ 2 }A=1-{ Cos }^{ 2 }A$

$\displaystyle (2)\quad 1+{ tan }^{ 2 }A={ Sec }^{ 2 }A\\ \Rightarrow { Sec }^{ 2 }A-{ tan }^{ 2 }A=1$

$\displaystyle (3)\quad 1+{ cot }^{ 2 }A={ cosec }^{ 2 }A\\ \Rightarrow { cosec }^{ 2 }A-{ cot }^{ 2 }A=1$

$\displaystyle (4)\quad \tan { A } =\frac { \sin { A } }{ \cos { A } } \quad and\quad \cot { A } =\frac { \cos { A } }{ \sin { A } }$

(5) Fundamental inequalities: For 0<A<π /2

$\displaystyle 0<\cos { A } <\frac { \sin { A } }{ A } <\frac { 1 }{ \cos { A } }$

(6) It is possible to express trigonometrical ratios in terms of any one of them as,

$\displaystyle \sin { \theta } =\frac { 1 }{ \sqrt { 1+{ cot }^{ 2 }\theta } } \\ \cos { \theta } =\frac { { cot }^{ 2 }\theta }{ \sqrt { 1+{ cot }^{ 2 }\theta } } \\ \tan { \theta } =\frac { 1 }{ { cot }\theta } \\ cosec\theta =\sqrt { 1+{ cot }^{ 2 }\theta } \\ \sec { \theta } =\frac { \sqrt { 1+{ cot }^{ 2 }\theta } }{ { cot }\theta }$

February 22, 2019