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Mathematics Lab Manual Class XII | 14 Activities

    Mathematics Lab Manual Class XII 14 lab activities for class 12 with complete observation Tables strictly according to the CBSE syllabus also very useful & helpful for the students and teachers. General instructions All these activities are strictly according to the CBSE syllabus. Students need to complete atleast 12 activity from the list of 14 activities. Students can make their own selection.

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
Radian Measure:- Angle made by an arc of unit length in a circle of unit radius is called one radian
\[One\; Radian =\frac{Unit\; length\; of\; arc}{Unit\; Radius}\]\[\theta \; Radian =\frac{length\; of\; arc}{Radius\; of\; Circle}\; \; or\: \: \theta =\frac{l}{r}\; \; or\; \; l=r\theta\]
RELATION BETWEEN DEGREE AND RADIAN
\[Angle\; at\; the\; centre\; of\; the\; circle =2\pi \; or\; 360^{o}\]\[2\pi \; Radian=360^{o}\Rightarrow \; \; \pi \; Radian=180^{o}\]\[1\: Radian=\frac{180}{\pi }\; \; \frac{180\times 7}{22}=57^{o}16^{'}\]\[180^{o}=\pi Radian\; \; \Rightarrow \; 1^{o}=\pi Radian=\frac{22}{7\times 180}Radian=0.01746\; Radian\]
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
https://dinesh51.blogspot.com

SIGN OF TRIGONOMETRIC RATIOS IN DIFFERENT QUADRANT
https://dinesh51.blogspot.com
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 90(I and II Quadrant)
sin(90 - θ)
+cosθ

sin(90 + θ)
+cosθ
cos(90 – θ)
+Sinθ
cos(90 + θ)
-Sinθ
tan(90 - θ) 
+cotθ
tan(90 + θ) 
-cotθ
cot(90 - θ)
+tanθ
cot(90 + θ)
-tanθ
sec(90 - θ)
+cosecθ
sec(90 + θ)
-cosecθ
cosec(90 - θ)
+secθ
cosec(90 + θ)
+secθ

Trigonometric Transformations at 180o  (II and III Quadrant)
sin(180 - θ)
Sin θ

sin(180 + θ)
-Sin θ
cos(180 – θ)
-Cos θ
cos(180 + θ)
-Cos θ
tan(180 - θ) 
-tanθ
tan(180 + θ) 
+tanθ
cot(180 - θ)
-cotθ
cot(180 + θ)
+cotθ
sec(180 - θ)
-secθ
sec(180 + θ)
-secθ
cosec(180 - θ)
cosecθ
cosec(180 + θ)
-cosecθ

Trigonometric Transformations at 270o  (III and IV Quadrant)
sin(270 - θ)
-cosθ

sin(270 + θ)
-cosθ
cos(270 – θ)
-Sinθ
cos(270 + θ)
+Sinθ
tan(270 - θ) 
+cotθ
tan(270 + θ) 
-cotθ
cot(270 - θ)
+tanθ
cot(270 + θ)
-tanθ
sec(270 - θ)
-cosecθ
sec(270 + θ)
+cosecθ
cosec(270 - θ)
-secθ
cosec(270 + θ)
-secθ

Trigonometric Transformations at 360o  (IV and I Quadrant)
sin(360 - θ)
-Sin θ

sin(360 + θ)
+Sin θ
cos(360 – θ)
+Cos θ
cos(360 + θ)
+Cos θ
tan(360 - θ) 
-tanθ
tan(360 + θ) 
+tanθ
cot(360 - θ)
-cotθ
cot(360 + θ)
+cotθ
sec(360 - θ)
+secθ
sec(360 + θ)
+secθ
cosec(360 - θ)
-cosecθ
cosec(360 + θ)
+cosecθ

Trigonometric Transformations at 0o   (IV and I Quadrant)
sin(0 - θ)
-Sin θ

sin(0+θ)
+ Sin θ
cos(0 - θ)
+Cos θ
cos(0+θ)
+ Cos θ
tan(0 - θ) 
-tanθ
tan(0+θ) 
+ tanθ
cot( 0-θ)
-cotθ
cot(0+θ)
+ cotθ
sec( 0-θ)
+secθ
sec(0+θ)
+ secθ
cosec( 0-θ)
-cosecθ
cosec(0+θ)
+ cosecθ

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)

THANKS FOR YOUR VISIT
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