कुछ विशिष्ट कोणों के त्रिकोणमितीय अनुपात

In this section, we will find the values of the trigonometric ratios for angles of $0^{\circ },30^{\circ },45^{\circ },60^{\circ },90^{\circ }$.

Trigonometric Ratios of 45°

Fig.1 Triangle

In $\bigtriangleup ABC$ right angled at $B$ , If $\angle A=45^{\circ }$, $\angle C=45^{\circ }$

$BC=AB=a$

Using Pythagoras Theorem

$AB^{2}+BC^{2}=AC^{2}$

$a^{2}+a^{2}=2a^{2}$

$AC=a{\sqrt {2}}$

$sin\ 45^{\circ }={\frac {side\ opposite\ to\ angle\ 45^{\circ }}{hypotenuse}}={\frac {BC}{AC}}={\frac {a}{a{\sqrt {2}}}}={\frac {1}{\sqrt {2}}}$

$cos\ 45^{\circ }={\frac {side\ adjacent\ to\ angle\ 45^{\circ }}{hypotenuse}}={\frac {AB}{AC}}={\frac {a}{a{\sqrt {2}}}}={\frac {1}{\sqrt {2}}}$

$tan\ 45^{\circ }={\frac {side\ opposite\ to\ angle\ 45^{\circ }}{side\ adjacent\ to\ angle\ 45^{\circ }}}={\frac {BC}{AB}}={\frac {a}{a}}=1$

$cosec\ 45^{\circ }={\frac {1}{sin\ 45^{\circ }}}={\sqrt {2}}$ , $sec\ 45^{\circ }={\frac {1}{cos\ 45^{\circ }}}={\sqrt {2}}$ , $cot\ 45^{\circ }={\frac {1}{tan\ 45^{\circ }}}=1$

Trigonometric Ratios of 30° and 60°

Fig. 2 Triangle

Consider an equilateral $\bigtriangleup ABC$. Each angle in an equilateral triangle is $60^{\circ }$, therefore,$\angle A=\angle B=\angle C=60^{\circ }$ .

Draw a perpendicular $AD$ from $A$ to the side $BC$ (see Fig. 2).

Now $\bigtriangleup ABD=\bigtriangleup ACD$

Therefore, $BD=DC$ and $\angle BAD=\angle CAD$ (Corresponding Parts of Congruent Triangles)

$\bigtriangleup ABD$ is a right angled triangle , right angled at $D$ with $\angle BAD=30^{\circ }$ and $\angle ABD=60^{\circ }$

Let $AB=2a$ , Hence $BC=AC=AB=2a$

$BD={\frac {1}{2}}BC={\frac {1}{2}}\times 2a=a$

$AD^{2}=AB^{2}-BD^{2}=(2a)^{2}-a^{2}=3a^{2}$

$AD=a{\sqrt {3}}$

$sin\ 30^{\circ }=\ {\frac {BD}{AB}}={\frac {a}{2a}}={\frac {1}{2}}$ , $cos\ 30^{\circ }=\ {\frac {AD}{AB}}={\frac {a{\sqrt {3}}}{2a}}={\frac {\sqrt {3}}{2}}$ , $tan\ 30^{\circ }=\ {\frac {BD}{AD}}={\frac {a}{a{\sqrt {3}}}}={\frac {1}{\sqrt {3}}}$

$cosec\ 30^{\circ }={\frac {1}{sin\ 30^{\circ }}}=2$ , $sec\ 30^{\circ }={\frac {1}{cos\ 30^{\circ }}}={\frac {2}{\sqrt {3}}}$ , $cot\ 30^{\circ }={\frac {1}{tan\ 30^{\circ }}}={\sqrt {3}}$

Similarly

$sin\ 60^{\circ }=\ {\frac {AD}{AB}}={\frac {a{\sqrt {3}}}{2a}}={\frac {\sqrt {3}}{2}}$ , $cos\ 60^{\circ }=\ {\frac {BD}{AB}}={\frac {a}{2a}}={\frac {1}{2}}$ , $tan\ 60^{\circ }=\ {\frac {AD}{BD}}={\frac {a{\sqrt {3}}}{a}}={\sqrt {3}}$

$cosec\ 60^{\circ }={\frac {2}{\sqrt {3}}}$ , $sec\ 60^{\circ }={\frac {1}{cos\ 60^{\circ }}}=2$ , $cot\ 60^{\circ }={\frac {1}{tan\ 60^{\circ }}}={\frac {1}{\sqrt {3}}}$

Trigonometric ratios of 0°, 30°, 45°, 60° and 90°

$\angle A$	$0^{\circ }$	$30^{\circ }$	$45^{\circ }$	$60^{\circ }$	$90^{\circ }$
$sin\ A$	$0$	${\frac {1}{2}}$	${\frac {1}{\sqrt {2}}}$	${\frac {\sqrt {3}}{2}}$	$1$
$cos\ A$	$1$	${\frac {\sqrt {3}}{2}}$	${\frac {1}{\sqrt {2}}}$	${\frac {1}{2}}$	$0$
$tan\ A$	$0$	${\frac {1}{\sqrt {3}}}$	$1$	${\sqrt {3}}$	Not Defined
$cosec\ A$	Not Defined	$2$	${\sqrt {2}}$	${\frac {2}{\sqrt {3}}}$	$1$
$sec\ A$	$1$	${\frac {2}{\sqrt {3}}}$	${\sqrt {2}}$	$2$	Not Defined
$cot\ A$	Not Defined	${\sqrt {3}}$	$1$	${\frac {1}{\sqrt {3}}}$	$0$