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
\[1\; Right\; angle=90^{o}\]\[1\; degree=60\; minute\; \; or\: \: 1^{o}=60^{'}\]\[1\; minute=60\; second\; \; or\: \: 1^{'}=60^{''}\]
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
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 ANGLESTrigonometric Transformations at 90o (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
\[Sin\theta =2\: sin\: \frac{\theta }{2}\: cos\: \frac{\theta }{2},\:\: or\: \:\; \frac{2tan\frac{\theta }{2}}{1+tan^{2}\frac{\theta }{2}}\]\[Cos\theta =Cos^{2}\; \frac{\theta }{2} -Sin^{2}\;\frac{\theta }{2},\; \; or\; \; 2Cos^{2}\; \frac{\theta }{2} -1,\; \; or\; \; 1-2Sin^{2}\; \frac{\theta}{2} \; \; or\; \; \frac{1-tan^{2}\; \frac{\theta}{2} }{1+tan^{2}\; \frac{\theta}{2} }\]\[Tan\theta =\frac{2tan\frac{\theta }{2}}{1-tan^{2}\frac{\theta }{2}}\]
\[Sin^{2}\; \frac{\theta}{2} =\frac{1-cos\; \theta }{2},\: \:\; \; \: Cos^{2}\; \frac{\theta}{2} =\frac{1+cos\: \theta }{2}\]\[tan^{2}\; \frac{\theta}{2} =\frac{1-cos\theta }{1+cos\theta },\]
\[2SinA\: CosB=Sin(A+B)+Sin(A-B)\]\[2CosA\: SinB=Sin(A+B)-Sin(A-B)\]\[2CosA\: CosB=Cos(A+B)+Cos(A-B)\]\[2SinA\: SinB=-Cos(A+B)+Cos(A-B)\]
\[SinC+SinD=2Sin\left (\frac{C+D}{2} \right )Cos\left ( \frac{C-D}{2} \right )\]\[SinC-SinD=2Cos\left (\frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )\]\[CosC+CosD =2Cos\left (\frac{C+D}{2} \right )Cos\left ( \frac{C-D}{2} \right )\]\[CosC-CosD=-2Sin\left (\frac{C+D}{2} \right )Sin\left ( \frac{C-D}{2} \right )\]
\[Tan\left ( 45+\theta \right )=\frac{1+tan\theta }{1-tan\theta }\]\[Tan\left ( 45-\theta\right )=\frac{1-tan\theta }{1+tan\theta }\]\[Sin^{2}A-Sin^{2}B=Sin(A+B)Sin(A-B)\]\[Cos^{2}A-Cos^{2}B=Sin(A+B)Sin(A-B)\]
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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