Trigonometry

1. If P(x,y) is any point in the cartesian plane. Sin$\theta$ = $\frac{y}{r}$ , cos$\theta$ = $\frac{x}{r}$, tan $\theta$= $\frac{y}{x}$, where x $\ne$ 0 and $x^2$ + $y^2$= $r^2$ where r$>$0.
2. Relation between trignometric ratios:-

(i) ${\sin }^2$$\theta$ + ${\cos }^2$$\theta$= 1$\Rightarrow$ ${\sin }^2$$\theta$= 1 - ${\cos }^2$$\theta$, ${\cos }^2$$\theta$= 1 - ${\sin }^2$$\theta$

(ii) ${\sec }^2$$\theta$ - ${\tan }^2$$\theta$= 1$\Rightarrow$ ${\sec }^2$$\theta$ = 1 + ${\tan }^2$$\theta$, ${\tan }^2$$\theta$ = ${\sec }^2$$\theta$ - 1

(iii) ${\mathrm{cosec}}^2$$\theta$ - ${\cot }^2$$\theta$ = 1$\Rightarrow$ ${\mathrm{cosec}}^2$$\theta$= 1 + ${\cot }^2$$\theta$, ${\cot }^2$$\theta$= ${\mathrm{cosec}}^2$$\theta$-1

3. (i) tan $\theta$= $\frac{\sin \theta }{\cos \theta }$

4. (i) cosec$\theta$= $\frac{1}{\sin \theta }$

(ii) sec$\theta$= $\frac{1}{\cos \theta }$

(iii) cot$\theta$= $\frac{1}{\tan \theta }$

5. Relation between degree measure and Radian measure:-

$\pi$ radians= ${180}^0$ $\Rightarrow$ $\frac{\pi }{2}$= ${90}^0$, $\frac{\pi }{3}$= ${60}^0$, $\frac{\pi }{4}$= ${45}^0$, $\frac{\pi }{6}$= ${30}^0$, $\frac{\pi }{10}$= ${18}^0$, $\frac{\pi }{5}$= ${36}^0$, $2\frac{\pi }{5}$= ${72}^0$

6. Table of T-ratios of standard angles

7. Signs of Trignometric functions

8. General reduction formula:-

The given angle can be put in the form $(\frac{n\pi }{2}\pm \theta )$

1. If 'n' is even same function exists.

2. If 'n' is odd, co-function exists and the sign of the function is according to which quadrant the given function and the given angle lie.

Addition and Subtraction formulae

9. sin (A+B) = sin A cosB + cosA sinB

sin(A-B)= sinA cosB - cosA sinB

cos(A+B)= cosA cosB - sinA sinB

cos(A-B)= cosA cosB + sinA sinB

tan (A+B)= $\frac{\tan A+\mathrm{tanB}}{1-\mathrm{tanA}\mathrm{tanB}}$

tan (A-B) = $\frac{\mathrm{tanA}-\mathrm{tanB}}{1+\mathrm{tanA}\mathrm{tanB}}$

10. tan (A+B+C) = $\frac{\mathrm{tanA}+\mathrm{tanB}+\mathrm{tanC}-\mathrm{tanA}\mathrm{tanB}\mathrm{tanC}}{1-(\mathrm{tanA}\mathrm{tanB}+\mathrm{tanB}\mathrm{tanC}+\mathrm{tanC}\mathrm{tanA}}$

11. If A+B+C= $\pi$

(i)tanA + tanB+ tanC= tanA. tanB. tanC

(ii) cotA cotB+ cotB cotC + cotC cotA= 1

(iii) tan$\frac{A}{2}$ tan$\frac{B}{2}$ + tan$\frac{B}{2}$ tan$\frac{C}{2}$+ tan $\frac{C}{2}$tan $\frac{A}{2}$= 1

(iv) cot$\frac{A}{2}$ + cot $\frac{B}{2}$ + cot $\frac{C}{2}$= cot $\frac{A}{2}$. cot $\frac{B}{2}$. cot $\frac{C}{2}$

Sin (A+B+C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC - sinA sinB sinC

Cos(A+B+C)= cosA cosB cosC - cosA sinB sinC - sinA cosB sinC - sinA sinB cosC

Multiple and sub-multiple angles formulae

12. sin 2A= $\{\frac{\frac{2\mathrm{sinA}\mathrm{cosA}}{2\mathrm{tanA}}}{1+{\tan }^{2}A}$

cos 2A= $\{{\cos }^{2}A-{\sin }^{2}A$

${2\cos }^2$A - 1

1 - 2${\sin }^2$A

$\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}$

13. 1+cos2A=2${\cos }^2$A$\Rightarrow$ 1+ cos A = 2${\cos }^2(A/2)$

1-cos2A=2${\sin }^2$A$\Rightarrow$ 1- cos A = 2${\sin }^2(A/2)$

$\frac{1-\cos 2A}{1+\cos 2A}$ = ${\tan }^{2}A$

14. SinA=$\{\frac{\underset{2\tan A/2}{2\sin A/2\cos A/2}}{1+{\tan }^2(A/2)}$

Cos A= $\{\frac{\underset{1-{\tan }^{2}A/2}{\underset{1-2{\sin }^{2}A/2}{\underset{2{\cos }^{2}A/2}{{\mathrm{Cos}}^{2}A/2-{\sin }^{2}A/2}}}}{1+{\tan }^{2}A/2}$

15. sin3A= 3sinA - 4${\sin }^3$A

cos3A= 4${\cos }^3$A - 3 cosA

${\sin }^3$A= $\frac{3\mathrm{sinA}-\sin 3A}{4}$

${\cos }^3$A = $\frac{3\cos A+\cos 3A}{4}$

${\tan }^3$A= $\frac{3\mathrm{tanA}-{\tan }^{3}A}{1-3{\tan }^{2}A}$

16. sin (A+B) sin(A-B) = ${\sin }^2$A- ${\sin }^2$B = ${\cos }^2$B - ${\cos }^2$A

cos(A+B) cos(A-B) = ${\cos }^2$A - ${\sin }^2$B = ${\cos }^2$B - ${\sin }^2$A

17. sin ($\pi$ / 10) = sin ${18}^0$= $\frac{\sqrt{5}-1}{4}$

cos ($\pi$ / 10) = cos${18}^0$= $\frac{\sqrt{10+2\sqrt{5}}}{4}$

sin${75}^0$= cos${15}^0$= $\frac{\sqrt{3}+1}{2\sqrt{2}}$

cos${75}^0$= sin${15}^0$= $\frac{\sqrt{3}+1}{2\sqrt{2}}$

cos ${75}^0$= sin${15}^0$= $\frac{\sqrt{3}-1}{2\sqrt{2}}$

sin $\frac{\pi }{5}$= sin${36}^0$= m$\frac{\sqrt{10-2\sqrt{5}}}{4}$

cos $\frac{\pi }{5}$= cos${36}^0$= $\frac{\sqrt{5}+1}{4}$

sin${54}^0$= sin (${90}^0$
- ${36}^0$) = cos${36}^0$

cos${54}^0$ = cos(${90}^0$ - ${36}^0$)= sin${36}^0$

sin${72}^0$ = sin (${90}^0$- ${18}^0$)= cos${18}^0$

cos${72}^0$ = cos(${90}^0$ - 18) = sin ${18}^0$

18. sin ($\pi$/8) = sin ${22}^{{1/2}^0}$= $\frac{\sqrt{2-\sqrt{2}}}{2}$

cos ($\pi$/8)= cos ${22}^{{1/2}^0}$= $\frac{\sqrt{2+\sqrt{2}}}{2}$

tan ($\pi$/8)= tan ${22}^{{1/2}^0}$= $\sqrt{2}-1$

Sum (Difference) $\to$ Product formulae

19. sin C + sin D= 2 sin $\frac{C+D}{2}$ cos $\frac{C-D}{2}$

sin C - sin D= 2cos $\frac{C+D}{2}$ sin $\frac{C-D}{2}$

cos C + cos D= 2cos $\frac{C+D}{2}$ cos $\frac{C-D}{2}$

cos C - cos D= -2sin $\frac{C+D}{2}\sin \frac{C-D}{2}$

or cosD - cosC= 2sin $\frac{C+D}{2}$ sin $\frac{C-D}{2}$

20. Special results

(i) CosA. cos2A. cos4A....... cos($2^{n-1}$ A)= $\frac{1}{2^n\sin A}$ sin$\left(2^{n}A\right)$

(ii) cosA + cos(A+B) + cos (A+2B) +....... n terms = $\frac{\sin \mathrm{nB}/2}{\sin B/2}$ $\times$ cos $\left(\frac{\mathrm{first}\mathrm{angle}+\mathrm{last}\mathrm{angle}}{2}\right)$

(iii) sinA + sin(A+B) + sin(A+2B)+........ + n terms = $\frac{\sin nB/2}{\sin B/2}$ $\times$ sin $\left(\frac{\mathrm{first}\mathrm{angle}+\mathrm{last}\mathrm{angle}}{2}\right)$

(iv) tan ($x_1$
+ $x_2$ + $x_3$+...........+ $x_n$) = $\frac{S_1-S_3+S_5-\mathrm{..........}}{1-S_2+S_4-S_6\mathrm{........}}$

where $S_r$ stands for the sum of the products of tan $x_1$, tan$x_2$,......tan$x_n$ taken 'r' at a

time.

(v) ${\sin }^4$A + ${\cos }^4$A= 1 - 2${\sin }^2$A ${\cos }^2$A

(vi) ${\sin }^6$A + ${\cos }^6$A= 1 - 3${\sin }^2$A ${\cos }^2$A

(vii) (cos A + cosB) (cos2A + cos 2B)..........

(cos $2^n$A + cos $2^n$B)= $\frac{\cos 2^{n+1}A-\cos 2^{n+1}\mathrm{.}B}{2^n(\cos A-\mathrm{cosB})}$


Product$\to$ Sum (Difference) formulae


21. sin A cos B= 1/2 [ sin (A+B) + sin (A-B)]

cos A sin B= 1/2 [sin (A+B) - sin (A-B)]

cos A cos B= 1/2 [cos (A+B) + cos (A-B)]

sin A sin B= 1/2 [cos (A+B) - cos (A-B)]

22. Periodic Function: f(x) is periodic with period t if t is the least positive integer such that f(x+t) = f(x)

eg: f(x) = sinx, f(x+2$\pi$) = sin (2$\pi$+x) = sin x= f(x)

sin x is periodic with period 2$\pi$ and tan x is periodic with period $\pi$.
Note: 1.

If $f_1$(x) and $f_2$(x) are periodic having the same period t, then a $f_1$(x) + b $f_2$(x) is also periodic having the same period t.
Note: 2.

If t is the fundamental period of the periodic function f(x) then f(ax+b) where a$>$0 and b any number is also a periodic function with fundamental period = t/a.

eg: f(x) = sin x is periodic having period 2$\pi$.

f(x) = sin 3x is periodic having period 2$\pi$/3

23.

Arc AB= l = r$\theta$ where $\theta$ is measured in radians

Area of sector AOB is a= $\frac{1}{2}$ $r^2$ $\theta$

24. Trigonometric Equations:

1. sin x= 0 $\Rightarrow$ x= n$\pi$, n$\in$I

2. cos x= 0 $\Rightarrow$ x= (2n +1) $\frac{\pi }{2}$, n$\in$I

3. tan x= 0 $\Rightarrow$ x= n$\pi$, n$\in$I

4. sin x= sin$\alpha$ $\Rightarrow$ x= n$\pi$ + ${(-1)}^n$ $\alpha$, n$\in$I

5. cos x= cos$\alpha$ $\Rightarrow$ x = 2$\bigcap$ $\pi$ $\pm$ $\alpha$, n $\in$I

6. tan x= tan$\alpha$ $\Rightarrow$ x= n$\pi$ + $\alpha$, n$\in$I

7. ${\sin }^2$x= ${\sin }^2$ $\alpha$ $\Rightarrow$ x= n$\pi$ + $\alpha$, n$\in$I

8. ${\cos }^2$x= ${\cos }^2$ $\alpha$ $\Rightarrow$ x= n$\pi$ $\pm$ $\alpha$, n$\in$I

9. ${\tan }^2$x= ${\tan }^2$ $\alpha$ $\Rightarrow$ x= n$\pi$ $\pm$ $\alpha$, n$\in$I

10. If sin$\theta$= 1, $\theta$= n$\pi$+ ${(-1)}^n$ $\pi$/2 or 2n$\pi$ + ($\pi$/2). If cos$\theta$ = +1, $\theta$= 2n$\pi$

11. If sin$\theta$= -1, $\theta$= n$\pi$ - ${(-1)}^n$ $\pi$/2 or 2n$\pi$ - ($\pi$/2)

12. If cos$\theta$ = -1, $\theta$= (2n+1)$\pi$

13. If cos$\theta$= 0, $\theta$ = 2n$\pi \pm$ $\pi$/2 or n$\pi$ + $\pi$/2

14. To solve a cos $\theta$ + b sin $\theta$ = c put a = r cos $\alpha$, b= r sin $\alpha$

Note:1

If sin$\theta$ = sin$\alpha$, cos $\theta$ = cos$\alpha$, then $\theta$ = 2n$\pi$ + $\alpha$
Note:2

If ${\theta }_1$ is a solution of sin$\theta$= k where |K|$\le$1 then ${-\theta }_1$, is a solution of sin$\theta$= -k

If ${\theta }_1$ is a solution of cos$\theta$ = k where |K|$\le$1 then ${-\theta }_1$, is also a solution of cos$\theta$= -k

If ${\theta }_1$ is a solution of tan$\theta$ = k where k$\in$R then ${-\theta }_1$, is a solution of tan$\theta$= -k

1. Inverse Trigonometric Functions:

If sin$\theta$= x, the valve of $\theta$ lying in [-$\pi$/2, $\pi$/2] is called the principal valve of ${\sin }^{-1}$x and is denoted by ${\sin }^{-1}$x or arc sin x.

ie ${\sin }^{-1}$x $\in$ [-$\pi$/2, $\pi$/2]

If cos$\theta$= x, the valve of $\theta$ lying in [0, $\pi$] is called the principal valve of ${\cos }^{-1}$x and is denoted by ${\cos }^{-1}$x or arc cosx.

ie, arc cos x $\in$ [0, $\pi$]

If tan$\theta$= x, the valve of $\theta$ lying in [-$\pi$/2, $\pi$/2] is called the principal valve of ${\tan }^{-1}$x and is denoted by ${\tan }^{-1}$x or arc tanx.

ie, ${\tan }^{-1}$x $\in$ [-$\pi$/2, $\pi$/2]

2. Principal valves of inverse trignometric functions

Note:

${\sin }^{-1}$(1/2)= $\pi$/6 not 5$\pi$/6

${\cos }^{-1}$ (-1/2)= 2$\pi$/3 not 4$\pi$/3

${\tan }^{-1}$ (-$\sqrt{3}$)= -$\pi$/3 not 2$\pi$/3

${\cot }^{-1}$ (-1) = 3$\pi$ /4 not -$\pi$/4

3.

4.

5.

6.

${\sin }^{-1}x+{\sin }^{-1}$ y= $\pi$ if x = y = 1
${\sin }^{-1}x+{\sin }^{-1}$ y= $-\pi$ if x = y = -1
${\cos }^{-1}x+{\cos }^{-1}$ y= $0$ if x = y = 1
${\cos }^{-1}x+{\cos }^{-1}$ y= 2$\pi$ if x = y = -1

${\sin }^{-1}$ $\left(\frac{2x}{1+x^2}\right)$= 2 ${\cot }^{-1}$x = 2 ${\tan }^{-1}$(1/x), x$\ge$1

7.

8. ${\tan }^{-1}$x + ${\tan }^{-1}$y= ${\tan }^{-1}$ $\left[\frac{x+y}{1-\mathrm{xy}}\right]$, xy $<$ 1

= $\pi$ + ${\tan }^{-1}$ $\left[\frac{x+y}{1-\mathrm{xy}}\right]$ if xy $>$ 1
${\tan }^{-1}$x - ${\tan }^{-1}$y= ${\tan }^{-1}$$\left[\frac{x-y}{1+\mathrm{xy}}\right]$,

9. 2 ${\tan }^{-1}$x = ${\tan }^{-1}$ $\left[\frac{2x}{1-x^2}\right]$

= ${\sin }^{-1}$ $\left[\frac{2x}{1+x^2}\right]$

= ${\cos }^{-1}$ $\left[\frac{1-x^2}{1+x^2}\right]$

10. 2${\sin }^{-1}$ x= ${\sin }^{-1}$ $\left(2x\mathrm{.}\sqrt{1-x^2}\right)$, |x| $\le$ $\frac{1}{\sqrt{2}}$

= ${\cos }^{-1}$ ( 1- ${2x}^2$), x $\in$ [0, 1]
2 ${\cos }^{-1}$ x = ${\cos }^{-1}$ (${2x}^2$ - 1), x$\in$ [0,1]

${\tan }^{-1}$ $\left(\frac{1+x}{1-x}\right)$= $\frac{\pi }{4}$+ ${\tan }^{-1}$x, x $<$ 1

${\tan }^{-1}$ $\left(\frac{1-x}{1+x}\right)$= $\frac{\pi }{4}$- ${\tan }^{-1}$x, x $>$ -1

sin (${\cos }^{-1}$x)= cos(${\sin }^{-1}$x) = $\sqrt{1-x^2}$, |x| $\le$ 1

sec (${\mathrm{cosec}}^{-1}$x) = cosec (${\sec }^{-1}$x)= $\frac{|x|}{\sqrt{x^2-1}}$ |x| $>$1
tan (${\cot }^{-1}$x)= cot (${\tan }^{-1}$x) = $\frac{1}{x}$, x $>$0

11. 3 ${\sin }^{-1}$x= ${\sin }^{-1}$( 3x - $x^3$)

3 ${\cos }^{-1}$x= ${\cos }^{-1}$(${4x}^3$- 3x)

12. 3 ${\tan }^{-1}$(x) = ${\tan }^{-1}$ $\left[\frac{3x-x^3}{1-{3x}^2}\right]$

13. ${\sin }^{-1}$x + ${\sin }^{-1}$y = ${\sin }^{-1}$ $[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$

${\cos }^{-1}$x + ${\cos }^{-1}$y= ${\cos }^{-1}$ $[\mathrm{xy}-\sqrt{1-x^2}\mathrm{.}\sqrt{1-y^2}]$

x,y $\in$ [-1, 1], and L.H.S lies in [0,$\pi$]

${\sin }^{-1}$(cos $\theta$)= ${\sin }^{-1}\left(\sin (\frac{\pi }{2}-\theta )\right)$= $\frac{\pi }{2}$- $\theta$

${\cos }^{-1}$(sin $\theta$) =${\cos }^{-1}\left(\cos (\frac{\pi }{2}-\theta )\right)$= $\frac{\pi }{2}$- $\theta$

15. ${\tan }^{-1}$x + ${\tan }^{-1}$ y + ${\tan }^{-1}$ Z= ${\tan }^{-1}$ $\left[\frac{x+y+z-\mathrm{xyz}}{1-\mathrm{xy}-\mathrm{yz}-\mathrm{zx}}\right]$

16. If A, B, C are angles of a triangle then sin (B+C) = sin A etc.

cos (B+C) = -cos A etc.

[since A+B+C= ${180}^0$]

Also sin $\frac{B+C}{2}$= cos $\frac{A}{2}$ etc.

17. If A+B+C = ${180}^0$, then

(i) sin 2A + sin 2B + sin 2C = 4 sinA sinB sinC

(ii) cos 2A + cos 2B + cos 2C= -1 -4 cos A cosB cosC

(iii) sin A+ sin B + sinC = 4 cos $\frac{A}{2}$ cos $\frac{B}{2}$. cos $\frac{C}{2}$

(iv) cosA + cosB + cosC = 1+ 4sin $\frac{A}{2}$ sin $\frac{B}{2}$. sin $\frac{C}{2}$

(v) tan A + tan B + tanC = tanA tanB tanC

(vi) cotB cotC + cotC cotA + cotA cotB= 1

(vii) cot $\frac{A}{2}$ + cot$\frac{B}{2}$ + cot $\frac{C}{2}$= cot $\frac{A}{2}$ cot $\frac{B}{2}$. cot $\frac{C}{2}$

Solutions of triangles

1. Law of sines: $\frac{a}{\mathrm{sinA}}$= $\frac{b}{\mathrm{sinB}}$= $\frac{c}{\mathrm{sinC}}$= 2R
2. Law of cosines: $a^2$=$b^2$ = $c^2$- 2bc cos A $\Rightarrow$ cos A = $\frac{b^2+c^2-a^2}{2\mathrm{bc}}$
3. Law of tangents: tan$\frac{B-C}{2}$ = $\frac{b-c}{b+c}$cot$\frac{A}{2}$
4. Projection formula: a = b cos C + c cos B
5. sin A/2 = $\sqrt{\frac{(s-b)(s-c)}{\mathrm{bc}}}$ where s= $\frac{a+b+c}{2}$
6. cos A/2 = $\sqrt{\frac{s(s-a)}{\mathrm{bc}}}$
7. tan A/2= $\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$
8. sin A= 2/bc $\sqrt{s(s-a)(s-b)(s-c)}$= $\frac{2\mathrm{\Delta }}{\mathrm{bc}}$
9. $\mathrm{\Delta }$= $\frac{1}{2}$bc sinA = $\sqrt{s(s-a)(s-b)(s-c)}$

= $\frac{\mathrm{abc}}{4R}$= 2$R^2$ sinA sinB sinC

= 4Rr cos A/2 cosB/2 cosC/2

10. r = $\frac{\mathrm{\Delta }}{S}$= (s-a) tan (A/2)= 4R sin (A/2) sin (B/2) sin (C/2)
11. $r_1$= $\frac{\mathrm{\Delta }}{s-a}$= s tan (A/2) =4R sin (A/2) cos (B/2) cos (C/2)
12. $r_2$= $\frac{\mathrm{\Delta }}{s-b}$= s tan (B/2)
13. $r_3$= $\frac{\mathrm{\Delta }}{s-c}$= s tan (C/2)
14. $\frac{1}{r^1}$+$\frac{1}{r^2}$+$\frac{1}{r^3}$= $\frac{1}{r}$, $r_1$, $r_2$, $r_3$ - r= 4R

$r_1$ $r_2$ +$r_2$ $r_3$+$r_3$ $r_1$= $S^2$,

$\frac{1}{r^2}$ +$\frac{1}{r_1^2}$+$\frac{1}{r_2^2}$+$\frac{1}{r_3^2}$= $\frac{a^2+b^2+c^2}{{\mathrm{\Delta }}^2}$

$\frac{1}{\mathrm{bc}}$ +$\frac{1}{\mathrm{ca}}$+$\frac{1}{\mathrm{ab}}$=$\frac{1}{2\mathrm{Rr}}$

15. If $p_1$, $p_2$, $p_3$ are the altitudes from A, B, C to the opposite sides of a triangle ABC then

$\mathrm{\Delta }$ = $\frac{1}{2}$ ${\mathrm{ap}}_1$= $\frac{1}{2}$ ${\mathrm{bp}}_2$= $\frac{1}{2}{\mathrm{cp}}_3$
16. In triangle ABC,

sin A + sin B + sin C= (2R sin A sin B sin C)/r = s/R

cos A + cos B + cos C= 1 + r/R

r= 4R sin (A/2) sin (B/2) sin (C/2)

s= 4R cos (A/2) cos (B/2) cos (C/2)

a cos A + b cos B + c cos C= 4R sin A sin B sin C

17. In any triangle ABC, if O is the orthocentre AO= 2R cos A, OD= 2R cosB cosC

18. (a) The distance between the circumcentre O and orthocentre H is given by

OH = R$\sqrt{1-8\cos A\cos B\cos C}$

(b) The distance between the circumcentre O and centroid G is given by

OG= $\frac{1}{3}$OH

= $\frac{1}{3}$R $\sqrt{1-8\cos A\cos B\cos C}$

(c) Again

HG= $\frac{2}{3}$ $\sqrt{1-8\cos A\cos B\cos C}$

(d) The distance between the circumcentre O and the incentre 1 of $\mathrm{\Delta }$ ABC given by

OI= R$\sqrt{1-8\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}$

(e) If $l_1$ is the centre of the escribed circle opposite to the angle B, then

${\mathrm{OI}}_1$ = R$\sqrt{1+8\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}$

similarly,

${\mathrm{OI}}_2$ = R$\sqrt{1+8\cos \frac{A}{2}\sin \frac{B}{2}\cos \frac{C}{2}}$

${\mathrm{OI}}_3$ = R$\sqrt{1+8\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}}$

19. (a) Each interior angle of a regular polygon of n sides = $\frac{2n-4}{n}$rt. angles

= $\left(\frac{2n-4}{n}\right)$ $\frac{\pi }{2}$ radians

(b) R= $\frac{a}{2}$ cosec $\left(\frac{\pi }{n}\right)$

r= $\frac{a}{2}$cot $\left(\frac{\pi }{n}\right)$

where a is the length of a side of the polygon and R is circumradius and r is the in-

radius of the polygon.

(c) Area of the regular polygon of n sides = $\frac{1}{4}$ $\pi a^2$ cot $\left(\frac{\pi }{n}\right)$

or = ${\pi r}^2$ tan $\left(\frac{\pi }{n}\right)$ or = $\frac{n{R}^2}{2}$ sin $\frac{2\pi }{n}$
20. sinA + sinB + sinC is Max. when A=B=C

cosA + cosB + cosC is Max. when A=B=C

tanA + tanB + tanC is Min. when A=B=C

cotA + cotB + cot C is Min. when A=B=C