Proof of Basic Trigonometric Identities

In $ \displaystyle \vartriangle ABC$,

$ \displaystyle \sin \theta =\frac{a}{c}\ \ \ \ \ \ \ \ \cos \theta =\frac{b}{c}\ \ \ \ \ \ \ \ \tan \theta =\frac{a}{b}$

$ \displaystyle \cot \theta =\frac{b}{a}\ \ \ \ \ \ \ \ \sec \theta =\frac{c}{b}\ \ \ \ \ \ \ \ \operatorname{cosec}\theta =\frac{c}{a}$

$ \displaystyle \tan \theta =\frac{a}{b}=\frac{{\displaystyle \frac{a}{c}}}{{\displaystyle \frac{b}{c}}}=\frac{{\sin \theta }}{{\cos \theta }}$

$ \displaystyle \cot \theta =\frac{b}{a}=\frac{{\displaystyle \frac{b}{c}}}{{\displaystyle \frac{a}{c}}}=\frac{{\cos \theta }}{{\sin \theta }}$

$ \displaystyle \cot \theta =\frac{b}{a}=\frac{{\displaystyle \frac{b}{b}}}{{\displaystyle \frac{a}{b}}}=\frac{1}{{\tan \theta }}$

$ \displaystyle \sec \theta =\frac{c}{b}=\frac{{\displaystyle \frac{c}{c}}}{{\displaystyle \frac{b}{c}}}=\frac{1}{{\cos \theta }}$

$ \displaystyle \operatorname{cosec}\theta =\frac{c}{a}=\frac{{\displaystyle \frac{c}{c}}}{{\displaystyle \frac{a}{c}}}=\frac{1}{{\sin \theta }}$

$ \displaystyle \begin{array}{l}\ \ \ \text{By Pythagoras'}\ \text{ theorem,}\\\\\ \ \ {{a}^{2}}+{{b}^{2}}={{c}^{2}}\end{array}$

$ \displaystyle \therefore \frac{{{{a}^{2}}}}{{{{c}^{2}}}}+\frac{{{{b}^{2}}}}{{{{c}^{2}}}}=1$

$ \displaystyle \therefore {{\left( {\frac{a}{c}} \right)}^{2}}+{{\left( {\frac{b}{c}} \right)}^{2}}=1$

$\displaystyle \therefore \ \begin{array}{|l|} \hline \displaystyle {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\ \hline\end{array}$

$ \displaystyle \ \ \ \text{Since}\ {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$

$ \displaystyle \ \ \ \frac{{{{{\sin }}^{2}}\theta }}{{{{{\cos }}^{2}}\theta }}+\frac{{{{{\cos }}^{2}}\theta }}{{{{{\cos }}^{2}}\theta }}=\frac{1}{{{{{\cos }}^{2}}\theta }}$

$\displaystyle \therefore \ \begin{array}{|l|} \hline \displaystyle {{\tan }^{2}}\theta +1={{\sec }^{2}}\theta \\ \hline\end{array}$

$ \displaystyle \ \ \ \text{Since}\ {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$

$ \displaystyle \ \ \ \frac{{{{{\sin }}^{2}}\theta }}{{{{{\sin }}^{2}}\theta }}+\frac{{{{{\cos }}^{2}}\theta }}{{{{{\sin }}^{2}}\theta }}=\frac{1}{{{{{\sin }}^{2}}\theta }}$

$\displaystyle \therefore \ \begin{array}{|l|} \hline \displaystyle 1+{{\cot }^{2}}\theta ={{\operatorname{cosec}}^{2}}\theta \\ \hline\end{array}$

Exercise (1) - No.3 Solutions

(a) Angles (Degree) in Cartesian Plane

(b) Angles (Radian) in Cartesian Plane

 

Dynamic Representation

Exercise (1) - No.2 Solutions

$\displaystyle \displaystyle \begin{array}{{|l|}}\hline \displaystyle 1\ \text{radian} = \frac{{180{}^\circ }}{\pi } \\ \hline \end{array}$

$\displaystyle \displaystyle \begin{array}{{|l|}}\hline \displaystyle \theta\ \text{radians} =\theta\times \frac{{180{}^\circ }}{\pi } \\ \hline \end{array}$

$ \displaystyle \text{(a)}\ 2\pi \ \text{rad}=\text{2}\pi \times \frac{{180{}^\circ }}{\pi }=360{}^\circ $

$ \displaystyle \text{(b)}\ \frac{{11\pi }}{6}\ \text{rad}=\frac{{11\pi }}{6}\times \frac{{180{}^\circ }}{\pi }=330{}^\circ $

$ \displaystyle \text{(c)}\ \frac{{7\pi }}{4}\ \text{rad}=\frac{{7\pi }}{4}\times \frac{{180{}^\circ }}{\pi }=315{}^\circ $

$ \displaystyle \text{(d)}\ \frac{{5\pi }}{3}\ \text{rad}=\frac{{5\pi }}{3}\times \frac{{180{}^\circ }}{\pi }=330{}^\circ $

$ \displaystyle \text{(e) }\frac{{3\pi }}{2}\ \text{rad}=\frac{{3\pi }}{2}\times \frac{{180{}^\circ }}{\pi }=270{}^\circ $

$ \displaystyle \text{(f) }\frac{{4\pi }}{3}\ \text{rad}=\frac{{4\pi }}{3}\times \frac{{180{}^\circ }}{\pi }=240{}^\circ $

$ \displaystyle \text{(g) }\frac{{5\pi }}{4}\ \text{rad}=\frac{{5\pi }}{4}\times \frac{{180{}^\circ }}{\pi }=225{}^\circ $

$ \displaystyle \text{(h) }\frac{{7\pi }}{6}\ \text{rad}=\frac{{7\pi }}{6}\times \frac{{180{}^\circ }}{\pi }=210{}^\circ $

$ \displaystyle \text{(i) }\pi \ \text{rad}=\pi \times \frac{{180{}^\circ }}{\pi }=180{}^\circ $

$ \displaystyle \text{(j) }\frac{{5\pi }}{6}\ \text{rad}=\frac{{5\pi }}{6}\times \frac{{180{}^\circ }}{\pi }=150{}^\circ $

$ \displaystyle \text{(k) }\frac{{3\pi }}{4}\ \text{rad}=\frac{{3\pi }}{4}\times \frac{{180{}^\circ }}{\pi }=135{}^\circ $

$ \displaystyle \text{(l) }\frac{{2\pi }}{3}\ \text{rad}=\frac{{2\pi }}{3}\times \frac{{180{}^\circ }}{\pi }=120{}^\circ $

$ \displaystyle \text{(m) }\frac{\pi }{2}\ \text{rad}=\frac{\pi }{2}\times \frac{{180{}^\circ }}{\pi }=90{}^\circ $

$ \displaystyle \text{(n) }\frac{\pi }{3}\ \text{rad}=\frac{\pi }{3}\times \frac{{180{}^\circ }}{\pi }=60{}^\circ $

$ \displaystyle \text{(o) }\frac{\pi }{4}\ \text{rad}=\frac{\pi }{4}\times \frac{{180{}^\circ }}{\pi }=45{}^\circ $

$ \displaystyle \text{(p )}\frac{\pi }{6}\ \text{rad}=\frac{\pi }{6}\times \frac{{180{}^\circ }}{\pi }=30{}^\circ $
 

Positive and Negative Angles

Angles measured from the X-axis in an anticlockwise direction are positive angles.

Angles measured from the X-axis in a clockwise direction are negative angles.

Exercise (1) - No.1 Solutions

$\displaystyle \displaystyle\ \begin{array}{{|l|}}\hline \displaystyle 1{}^\circ =\frac{\pi }{{180}}\ \text{radians} \\ \hline \end{array}$

$\displaystyle \displaystyle \ \begin{array}{{|l|}}\hline \displaystyle \theta{}^\circ =\theta\times \frac{\pi }{{180}}\ \text{radians} \\ \hline \end{array}$  

$ \displaystyle \text{(a)}\ 30{}^\circ =30\times \frac{\pi }{{180}}=\frac{\pi }{6}\ \text{rad}$

$ \displaystyle \text{(b)}\ 45{}^\circ =45\times \frac{\pi }{{180}}=\frac{\pi }{4}\ \text{rad}$

$ \displaystyle \text{(c)}\ 60{}^\circ =60\times \frac{\pi }{{180}}=\frac{\pi }{3}\ \text{rad}$

$ \displaystyle \text{(d)}\ 90{}^\circ =90\times \frac{\pi }{{180}}=\frac{\pi }{2}\ \text{rad}$

$ \displaystyle \text{(e)}\ 120{}^\circ =120\times \frac{\pi }{{180}}=\frac{{2\pi }}{3}\ \text{rad}$

$ \displaystyle \text{(f)}\ 135{}^\circ =135\times \frac{\pi }{{180}}=\frac{{3\pi }}{4}\ \text{rad}$

$ \displaystyle \text{(g)}\ 150{}^\circ =150\times \frac{\pi }{{180}}=\frac{{5\pi }}{6}\ \text{rad}$

$ \displaystyle \text{(h)}\ 180{}^\circ =180\times \frac{\pi }{{180}}=\pi \ \text{rad}$

$ \displaystyle \text{(i)}\ 210{}^\circ =210\times \frac{\pi }{{180}}=\frac{{7\pi }}{6}\ \text{rad}$

$ \displaystyle \text{(j)}\ 225{}^\circ =225\times \frac{\pi }{{180}}=\frac{{5\pi }}{4}\ \text{rad}$

$ \displaystyle \text{(k)}\ 240{}^\circ =240\times \frac{\pi }{{180}}=\frac{{4\pi }}{3}\ \text{rad}$

$ \displaystyle \text{(l)}\ 270{}^\circ =270\times \frac{\pi }{{180}}=\frac{{3\pi }}{2}\ \text{rad}$

$ \displaystyle \text{(m)}\ 300{}^\circ =300\times \frac{\pi }{{180}}=\frac{{5\pi }}{3}\ \text{rad}$

$ \displaystyle \text{(n)}\ 315{}^\circ =315\times \frac{\pi }{{180}}=\frac{{7\pi }}{4}\ \text{rad}$

$ \displaystyle \text{(o)}\ 330{}^\circ =330\times \frac{\pi }{{180}}=\frac{{11\pi }}{6}\ \text{rad}$

$ \displaystyle \text{(p)}\ 360{}^\circ =360\times \frac{\pi }{{180}}=2\pi \ \text{rad}$

Degree-Radian Converter

Degree မွ Radian သို႕ ေျပာင္းရာတြင္ Degree တန္ဖိုးေနရာတြင္ ႏွစ္သက္ရာ ကိန္းျပည့္ တန္ဖိုး ရိုက္ထည့္ပါ။ 

Radian  မွ Degree  သို႕ ေျပာင္းရာတြင္ ပိုင္းေ၀ တန္ဖိုးေနရာတြင္ ႏွစ္သက္ရာ ကိန္းျပည့္ တန္ဖိုး  ရိုက္ထည့္ပါ။ ပိုင္းေျခ တန္ဖိုးေနရာတြင္ ႏွစ္သက္ရာ အေပါင္းကိန္းျပည့္ တန္ဖိုး  ရိုက္ထည့္ပါ။ 

ထို႔ေနာက္ enter ေခါက္ပါ။ 

Relation between Degrees and Radians

The radian measure of an angle is the ratio of the length of the arc (s) whose centre is at the vertex of the angle to the radius (r).

$\displaystyle \begin{array}{{|l|}}\hline \displaystyle \theta =\frac{s}{r} \\ \hline \end{array}$

For one complete anticlockwise revolution, the length of the arc is the length of entire circumference and hence $ \displaystyle s=2\pi r$. Therefore, the radian measure of one complete revolution is

$\displaystyle \begin{array}{{|l|}}\hline \displaystyle \theta =\frac{2\pi r}{r}{}=2\pi \ \text{radians} \\ \hline \end{array}$  

In degrees, one complete counterclockwise revolution is $\displaystyle 360°$ and hence $ \displaystyle \theta=360°$.

$ \displaystyle \begin{array}{l}\therefore \ \ 360{}^\circ =2\pi \ \text{radians}\\\\\therefore \ \ 180{}^\circ =\pi \ \text{radians}\\\\\therefore \ \ 1{}^\circ \times 180=\pi \ \text{radians}\\\end{array}$

$\displaystyle \therefore\ $ $\displaystyle \begin{array}{{|l|}}\hline \displaystyle 1{}^\circ =\frac{\pi }{{180}}\ \text{radians} \\ \hline \end{array}$

$ \displaystyle \begin{array}{l}\ \ \ \ \text{Since }\pi \ \text{radians}=180{}^\circ ,\\\\\ \ \ \ 1\ \text{radian}\ \times \pi =180{}^\circ \end{array}$

$\displaystyle \therefore\ $$\displaystyle \begin{array}{{|l|}}\hline \displaystyle 1\ \text{radian}\ =\frac{{180{}^\circ }}{\pi } \\ \hline \end{array}$

One Radian

The radian measure of an angle is the ratio of the length of the arc (s) whose centre is at the vertex of the angle to the radius (r).

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.

If s = r,

$ \displaystyle \text{central angle}=\frac{{\text{length}\ \text{of}\ \text{arc }AB}}{{\text{length}\ \text{of}\ \text{radius}}}=\frac{s}{r}=\frac{r}{r}=1\ \text{radian}$