Question

write all prove formula trigonometry

Answer 1

Definitions

Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.

The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:

{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}={\frac {a}{h}}} \sin \theta ={\frac  {{\mathrm  {opposite}}}{{\mathrm  {hypotenuse}}}}={\frac  {a}{h}}

{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}={\frac {b}{h}}} \cos \theta ={\frac  {{\mathrm  {adjacent}}}{{\mathrm  {hypotenuse}}}}={\frac  {b}{h}}

{\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {a}{b}}} \tan \theta ={\frac  {{\mathrm  {opposite}}}{{\mathrm  {adjacent}}}}={\frac  {a}{b}}

{\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}={\frac {b}{a}}} \cot \theta ={\frac  {{\mathrm  {adjacent}}}{{\mathrm  {opposite}}}}={\frac  {b}{a}}

{\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}={\frac {h}{b}}} \sec \theta ={\frac  {{\mathrm  {hypotenuse}}}{{\mathrm  {adjacent}}}}={\frac  {h}{b}}

{\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}={\frac {h}{a}}} \csc \theta ={\frac  {{\mathrm  {hypotenuse}}}{{\mathrm  {opposite}}}}={\frac  {h}{a}}

Answer 2

${ \red{ \bf{ Information \: related \: to \:Trigonometry:}}}$

${ \green{ \bf{ sin θ = Opposite Side/Hypotenuse }}}$

${ \green{ \bf{ cos θ = Adjacent Side/Hypotenuse }}}$

${ \green{ \bf{tan θ = Opposite Side/Adjacent Side }}}$

${ \green{ \bf{sec θ = Hypotenuse/Adjacent Side }}}$

${ \green{ \bf{ cosec θ = Hypotenuse/Opposite Side }}}$

${ \green{ \bf{ cot θ = Adjacent Side/Opposite Side }}}$

${ \red{ \bf{Their \: reciprocal \: Identities: }}}$

${ \green{ \bf{ cosec θ = 1/sin θ }}}$

${ \green{ \bf{ sec θ = 1/cos θ }}}$

${ \green{ \bf{ cot θ = 1/tan θ }}}$

${ \green{ \bf{sin θ = 1/cosec θ }}}$

${ \green{ \bf{ cos θ = 1/sec θ }}}$

${ \green{ \bf{ tan θ = 1/cot θ}}}$

${ \red{ \bf{ Their \: co-function \: Identities: }}}$

${ \green{ \bf{ sin (90°−x) = cos x }}}$

${ \green{ \bf{cos (90°−x) = sin x }}}$

${ \green{ \bf{ tan (90°−x) = cot x }}}$

${ \green{ \bf{ cot (90°−x) = tan x }}}$

${ \green{ \bf{ sec (90°−x) = cosec x }}}$

${ \green{ \bf{ cosec (90°−x) = sec x }}}$

${ \red{ \bf{ Their \: fundamental \: trigonometric \: identities: }}}$

${ \green{ \bf{ sin²θ + cos²θ = 1 }}}$

${ \green{ \bf{ sec²θ - tan²θ = 1 }}}$

${ \green{ \bf{ cosec²θ - cot²θ = 1 }}}$