### Math Assignment Class VIII | Square & Square Root

Math Assignment  Class VIII | Square & Square Root Download or Print free  assignment with answer key  for   Class  8 Squares and  Square Roots.   Important and extra questions that cover all topics of square and square root and is useful and helpful for the students. Math Assignment  Class VIII | Square & Square Root LEVEL -1

### Trigonometry Class 11 Chapter 3

TRIGONOMETRY : CLASS 11 : CHAPTER 3
Different systems of angles, radian measure, degree, AB and CD formulas,  transformations of angles in different quadrants and solution of trigonometric equations

Trigonometry:
It is the branch of mathematics in which we deal with the relation between the angle and sides of the right triangle.
Table 1
 0o 30o 45o 60o 90o 180o 270o 360o Sin 0 1/2 $1/\sqrt{2}$ $\sqrt{3}/2$ 1 0 -1 0 Cos 1 $\sqrt{3}/2$ $1/\sqrt{2}$ 1/2 0 -1 0 1 Tan 0 $1/\sqrt{3}$ 1 $\sqrt{3}$ $\infty$ 0 $\infty$ 0 Cot $\infty$ $\sqrt{3}$ 1 $1/\sqrt{3}$ 0 $\infty$ 0 $\infty$ Sec 1 $2/\sqrt{3}$ $\sqrt{2}$ 2 $\infty$ -1 $\infty$ 1 cosec $\infty$ 2 $\sqrt{2}$ $2/\sqrt{3}$ 1 $\infty$ -1 $\infty$

DIFFERENT SYSTEMS OF ANGLES
There are mainly three systems of measuring angles:
1) Centesimal System,  2) Sexagesimal System,  3) Circular System.

CENTESIMAL SYSTEM
$1\; Right angle =100\; grades\; \; or\: \: 90^{o}=100^{g}$$1\; grade=100\; minutes\; \; or\: \: 1^{g}=100^{'}$$1\; minute=100\; seconds\; \; or\: \: 1^{'}=100^{''}$
SEXAGESIMAL SYSTEM
$1\; Right\; angle=90^{o}$$1\; degree=60\; minute\; \; or\: \: 1^{o}=60^{'}$$1\; minute=60\; second\; \; or\: \: 1^{'}=60^{''}$
CIRCULAR SYSTEM
$One\; Radian =\frac{Unit\; length\; of\; arc}{Unit\; Radius}$$\theta \; Radian =\frac{length\; of\; arc}{Radius\; of\; Circle}\; \; or\: \: \theta =\frac{l}{r}\; \; or\; \; l=r\theta$
$Angle\; at\; the\; centre\; of\; the\; circle =2\pi \; or\; 360^{o}$$2\pi \; Radian=360^{o}\Rightarrow \; \; \pi \; Radian=180^{o}$$1\: Radian=\frac{180}{\pi }\; \; \frac{180\times 7}{22}=57^{o}16^{'}$$180^{o}=\pi Radian\; \; \Rightarrow \; 1^{o}=\pi Radian=\frac{22}{7\times 180}Radian=0.01746\; Radian$
Note : In order to convert degree measure into radian measure then we should multiply the angle in the degree measure by $\frac{\pi }{180}$
In order to convert a radian measure into a degree measure we should multiply the angle in the radian measure by $\frac{180 }{\pi }$
MOVEMENT OF ANGLE IN DIFFERENT  QUADRANT

SIGN OF TRIGONOMETRIC RATIOS IN DIFFERENT QUADRANT
TRIGONOMETRY-CBSE Mathematics
$For\: odd\: multiples\: of\: \frac{\pi }{2}\: i.e.\; \; \; \frac{\pi }{2}\pm \theta ,\: or\: \frac{3\pi }{2}\pm \theta \\sin\theta \leftrightharpoons cos\theta ,\: \: tan\theta \leftrightharpoons cot\theta ,\: \: sec\theta \leftrightharpoons cosec\theta$
COMPLETE  TRANSFORMATION OF ANGLES
 Trigonometric Transformations at 90o (I and II Quadrant) sin(90 - Î¸) +cosÎ¸ sin(90 + Î¸) +cosÎ¸ cos(90 – Î¸) +SinÎ¸ cos(90 + Î¸) -SinÎ¸ tan(90 - Î¸) +cotÎ¸ tan(90 + Î¸) -cotÎ¸ cot(90 - Î¸) +tanÎ¸ cot(90 + Î¸) -tanÎ¸ sec(90 - Î¸) +cosecÎ¸ sec(90 + Î¸) -cosecÎ¸ cosec(90 - Î¸) +secÎ¸ cosec(90 + Î¸) +secÎ¸

 Trigonometric Transformations at 180o  (II and III Quadrant) sin(180 - Î¸) Sin Î¸ sin(180 + Î¸) -Sin Î¸ cos(180 – Î¸) -Cos Î¸ cos(180 + Î¸) -Cos Î¸ tan(180 - Î¸) -tanÎ¸ tan(180 + Î¸) +tanÎ¸ cot(180 - Î¸) -cotÎ¸ cot(180 + Î¸) +cotÎ¸ sec(180 - Î¸) -secÎ¸ sec(180 + Î¸) -secÎ¸ cosec(180 - Î¸) cosecÎ¸ cosec(180 + Î¸) -cosecÎ¸

 Trigonometric Transformations at 270o  (III and IV Quadrant) sin(270 - Î¸) -cosÎ¸ sin(270 + Î¸) -cosÎ¸ cos(270 – Î¸) -SinÎ¸ cos(270 + Î¸) +SinÎ¸ tan(270 - Î¸) +cotÎ¸ tan(270 + Î¸) -cotÎ¸ cot(270 - Î¸) +tanÎ¸ cot(270 + Î¸) -tanÎ¸ sec(270 - Î¸) -cosecÎ¸ sec(270 + Î¸) +cosecÎ¸ cosec(270 - Î¸) -secÎ¸ cosec(270 + Î¸) -secÎ¸

 Trigonometric Transformations at 360o  (IV and I Quadrant) sin(360 - Î¸) -Sin Î¸ sin(360 + Î¸) +Sin Î¸ cos(360 – Î¸) +Cos Î¸ cos(360 + Î¸) +Cos Î¸ tan(360 - Î¸) -tanÎ¸ tan(360 + Î¸) +tanÎ¸ cot(360 - Î¸) -cotÎ¸ cot(360 + Î¸) +cotÎ¸ sec(360 - Î¸) +secÎ¸ sec(360 + Î¸) +secÎ¸ cosec(360 - Î¸) -cosecÎ¸ cosec(360 + Î¸) +cosecÎ¸

 Trigonometric Transformations at 0o   (IV and I Quadrant) sin(0 - Î¸) -Sin Î¸ sin(0+Î¸) + Sin Î¸ cos(0 - Î¸) +Cos Î¸ cos(0+Î¸) + Cos Î¸ tan(0 - Î¸) -tanÎ¸ tan(0+Î¸) + tanÎ¸ cot( 0-Î¸) -cotÎ¸ cot(0+Î¸) + cotÎ¸ sec( 0-Î¸) +secÎ¸ sec(0+Î¸) + secÎ¸ cosec( 0-Î¸) -cosecÎ¸ cosec(0+Î¸) + cosecÎ¸

STANDARD ANGLES IN DEGREE MEASURE AND RADIAN MEASURE
Table 2
 Degree Measure 30o 45o 60o 90o 180o 270o 360o Radian Measure $\frac{\pi }{6}$ $\frac{\pi }{4}$ $\frac{\pi }{3}$ $\frac{\pi }{2}$ $\pi$ $\frac{3\pi }{2}$ $2\pi$

TRIGONOMETRIC FORMULAS WITH COMPOUND ANGLE
Sin(A+B)=SinACosB + CosASinB
Sin(A-B)=SinACosB - CosASinB
Cos(A+B)=CosACosB - SinA SinB
Cos(A-B)=CosACosB + SinA SinB
$tan(A+B)=\frac{tanA+tanB}{1-tanA tanB}$$tan(A-B)=\frac{tanA-tanB}{1+tanA tanB}$$cot(A+B)=\frac{cotA cotB -1}{cotA + cotB}$$cot(A-B)=\frac{cotA cotB+1}{cotA - cotB}$
TRIGONOMETRIC FORMULAS WITH MULTIPLE ANGLE
$Sin2\theta =2Sin\theta\; Cos\theta \; \; \; OR\; \; \frac{2tan\theta }{1+tan^{2}\theta }$$Cos2\theta =Cos^{2}\theta -Sin^{2}\theta,\; \; or\: \: 2Cos^{2}\theta -1,\; \; or\; \; 1-2Sin^{2}\theta \; \; or\; \; \frac{1-tan^{2}\theta }{1+tan^{2}\theta }$$tan2\theta =\frac{2tan\theta }{1-tan^{2}\theta }$$Sin3\theta =3Sin\theta -4Sin^{3}\theta$$Cos3\theta =-\left ( 3Cos\theta -4Cos^{3}\theta \right )\: \: or\: \: \left ( 4Cos^{3}\theta -3Cos\theta \right )$$Tan3\theta =\frac{3tan\theta -tan^{3}\theta }{1-3tan^{2}\theta }$$Sin^{2}\theta =\frac{1-cos2\theta }{2},\: \: \: Sin^{2}4\theta =\frac{1-cos8\theta }{2}$$Cos^{2}\theta =\frac{1+cos\: 2\theta }{2},\: \: \: Cos^{2}\: 4\theta =\frac{1+cos\: 8\theta }{2}$$tan^{2}\; \theta =\frac{1-cos2\theta }{1+cos2\theta },\:\: \: \: \: tan^{2}\: 4\theta =\frac{1-cos\: 8\theta }{1+cos\: 8\theta }$
TRIGONOMETRIC FORMULAS WITH SUB-MULTIPLE ANGLE
$Sin\theta =2\: sin\: \frac{\theta }{2}\: cos\: \frac{\theta }{2},\:\: or\: \:\; \frac{2tan\frac{\theta }{2}}{1+tan^{2}\frac{\theta }{2}}$$Cos\theta =Cos^{2}\; \frac{\theta }{2} -Sin^{2}\;\frac{\theta }{2},\; \; or\; \; 2Cos^{2}\; \frac{\theta }{2} -1,\; \; or\; \; 1-2Sin^{2}\; \frac{\theta}{2} \; \; or\; \; \frac{1-tan^{2}\; \frac{\theta}{2} }{1+tan^{2}\; \frac{\theta}{2} }$$Tan\theta =\frac{2tan\frac{\theta }{2}}{1-tan^{2}\frac{\theta }{2}}$
$Sin^{2}\; \frac{\theta}{2} =\frac{1-cos\; \theta }{2},\: \:\; \; \: Cos^{2}\; \frac{\theta}{2} =\frac{1+cos\: \theta }{2}$$tan^{2}\; \frac{\theta}{2} =\frac{1-cos\theta }{1+cos\theta },$
A, B FORMULAS:-
$2SinA\: CosB=Sin(A+B)+Sin(A-B)$$2CosA\: SinB=Sin(A+B)-Sin(A-B)$$2CosA\: CosB=Cos(A+B)+Cos(A-B)$$2SinA\: SinB=-Cos(A+B)+Cos(A-B)$
C,D FORMULAS:-
$SinC+SinD=2Sin\left (\frac{C+D}{2} \right )Cos\left ( \frac{C-D}{2} \right )$$SinC-SinD=2Cos\left (\frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )$$CosC+CosD =2Cos\left (\frac{C+D}{2} \right )Cos\left ( \frac{C-D}{2} \right )$$CosC-CosD=-2Sin\left (\frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )$
SOME SPECIAL FORMULAS
$Tan\left ( 45+\theta \right )=\frac{1+tan\theta }{1-tan\theta }$$Tan\left ( 45-\theta\right )=\frac{1-tan\theta }{1+tan\theta }$$Sin^{2}A-Sin^{2}B=Sin(A+B)Sin(A-B)$$Cos^{2}A-Cos^{2}B=Sin(A+B)Sin(A-B)$
PERIODIC FUNCTIONS
All trigonometric ratios are periodic functions
$Sin(x+2\pi )=Sin(x+4\pi )=Sin(x+6\pi )=Sin(x+8\pi )=..........$$Cos(x+2\pi )=Cos(x+4\pi )=Cos(x+6\pi )=Cos(x+8\pi )=..........$$Sec(x+2\pi )=Sec(x+4\pi )=Sec(x+6\pi )=Sec(x+8\pi )=..........$$Cosec(x+2\pi )=Cosec(x+4\pi )=Cosec(x+6\pi )=Cosec(x+8\pi )=..........$$On\; changing\; x\; to\; 2n\pi,\; sinx,\; cosx,\; secx,\; cosecx\; all\; remain\; unchanged.$$sinx,\; cosx,\; secx,\; cosecx\; all\; are\; periodic\; functions\; and\; their\; period\; is\; 2\pi$$tanx=tan(x\pm \pi )=tan(x\pm 2\pi )=tan(x\pm 3\pi )=........$$cotx=cot(x\pm \pi )=cot(x\pm 2\pi )=cot(x\pm 3\pi )=........$$tanx\; and\; cotx\; are\; periodic\; and\; their\; period\; is\; \pi$
PRINCIPAL SOLUTION OF TRIGONOMETRIC FUNCTIONS
$Solutions\; of\; trigonometric\; equations\; for\; which\;\; 0\leq x\leq 2\pi \; \\ are\; called\; principal\; solutions$
PERIODIC VALUES:-
It is the least value which when added to the given function so that the value of that function remain unchanged.
TRIGONOMETRIC EQUATIONS:-
Equations involving trigonometric functions of a variable are called trigonometric equations.
GENERAL SOLUTIONS OF TRIGONOMETRIC EQUATIONS:-
$If\; sinx=0,\; \; then\; x=n\pi$$If\; cosx=0,\; \; then\; x=(2n+1)\frac{\pi }{2}$$If\; tanx=0,\; \; then\; x= n\pi$$If\; sinx=siny\; \; then\; x=n\pi +(-1)^{n}y$$If\; cosx=cosy,\; \; then\; x=2n\pi \pm y$$If\; tanx=tany,\; \; then\; x=n\pi +y$

Miscellaneous Exercise Chapter 3 Class 11
Q 8) If tanx = -4/3, and x lie in the II quadrant then find the value of sinx/2, cosx/2, tanx/2
Solution: Since angle lie in the II quadrant. Therefore
$\frac{\pi }{2}<x<\pi \Rightarrow \frac{\pi }{4}<\frac{x}{2}<\frac{\pi }{2}$⇒ Angle  x/2 lie in the first quadrant and in first quadrant sinx/2,  cosx/2 and tanx/2 all are positive  $tanx=\frac{-4}{3}=\frac{P}{B}$$\Rightarrow P= 4k,\: and\: B= -3k$$In\: \: \Delta OAB,\; OA=\sqrt{(4k)^{2}+(3k)^{2}}=5k$$Now\: \: cosx =\frac{B}{H}=\frac{-3k}{5k}=\frac{-3}{5}$$sin^{2}\; \frac{x}{2}=\frac{1-cosx}{2}=\frac{1-(\frac{-3}{5})}{2}$$sin^{2}\; \frac{x}{2}=\frac{1+\frac{3}{5}}{2}=\frac{8}{2\times 5}=\frac{4}{5}$$sin\; \frac{x}{2}=\sqrt{\frac{4}{5}}=\frac{2}{\sqrt{5}}$$cos^{2}\: \frac{x}{2}=\frac{1+cosx}{2}=\frac{1-\frac{3}{5}}{2}$$cos^{2}\: \frac{x}{2}=\frac{2}{2\times 5}=\frac{1}{5}$$cos\: \frac{x}{2}=\sqrt{\frac{1}{5}}=\frac{1}{\sqrt{5}}$$tan\frac{x}{2}=\frac{sin\frac{x}{2}}{cos\frac{x}{2}}=\frac{2/\sqrt{5}}{1/\sqrt{5}}=2$
 Worksheet(1) for the students   (Pair the following) Which do not have any pair write answer for them also 1) $tan2x$ a) $sinx$ 2) $\frac{tan2x}{1+tan^{2}x}$ b) $\frac{2tan\frac{x}{2}}{1-tan^{2}\frac{x}{2}}$ 3) $2cos^{2}\frac{x}{2}-1$ c) $\frac{2tan\frac{x}{2}}{1+tan^{2}\frac{x}{2}}$ 4) $sinx$ d) $\frac{2tanx}{1-tan^{2}x}$ 5) $\sqrt{1-cos^{2}x}$ e) $|sinx|$ 6) $\frac{1-tan\frac{x}{2}}{1+tan\frac{x}{2}}$ f)$sin2x$ g)$sinx$ 7) $sec^{2}x-1$ h)$cosx$ 8) $\frac{1-tan^{2}\frac{x}{2}}{1+tan^{2}\frac{x}{2}}$ i) $2sin\frac{x}{2}cos\frac{x}{2}$ j) $tan\left ( \frac{\pi }{4}-\frac{x}{2} \right )$ 9) $cosAcosB+sinAsinB$ k) $cos(B-A)$ 10) $tan\left (\frac{\pi }{4}-x \right )$ l)$cos(A+B)$ m) $\frac{1+tanx}{1-tanx}$

Answer Key : (1, d), (2, f), (3, n), (4, c and i), (5, e), (6, j), (7, tan2x), (8, h), (9, k), (10, n)

 Worksheet(2) for the students   Pair the following 1) $\sqrt{1-cosx}$ a$\frac{tanA-tanB}{1-tanAtanB}$ 2) $\sqrt{1+cosx}$ b $4cos^{3}x-3cosx$ 3) $\sqrt{1-sin2x}$ c $\frac{3tan-tan^{3}x}{1-3tan^{2}x}$ 4) $\sqrt{1+sin2x}$ d $sinAcosB-cosAsinB$ 5) $sin3x$ e $3sinx-4sin^{3}x$ 6) $cos3x$ f $|cosx-sinx|$ 7) $tan3x$ g $\left |\sqrt{2}sin^{2}\frac{x}{2} \right |$ 8) $sin(A-B)$ h $|cosx+sinx|$ 9) $tan(A-B)$ i $\left |\sqrt{2}cos^{2}\frac{x}{2} \right |$

Answer Key: (1, g), (2, i), (3, f), (4, h), 5, e), (6, b), (7, c), (8, d), (9, a)

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