Showing posts with label Trigonometry. Show all posts
Showing posts with label Trigonometry. Show all posts
Friday, December 22, 2017
CBSE Class XI - chap. I-3. Trigonometric Functions - Learning Companion
3. Trigonometric Functions
Positive and negative angles. Measuring angles in radians and in degrees and conversion of one into other.
Definition of trigonometric functions with the help of unit circle. Truth of the sin2x+cos2x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application. Deducing identities like the following:
Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x. General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.
Maths Trigonometry part 1 (Basic Concepts, Why Trigonometry) CBSE class 11 Mathematics XI
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ExamFear Education
Maths Trigonometry part 2 (Degree Measure) CBSE class 11 Mathematics XI
https://www.youtube.com/watch?v=1FijJ6tuqW4
Maths Trigonometry part 3 (Radian Measure) CBSE class 11 Mathematics XI
https://www.youtube.com/watch?v=Bleq2RyPFUE
Wednesday, December 24, 2014
33. Trigonometric ratios, Identities and Maximum & Minimum Values of Trigonometrical Expressions - Revision Facilitator
Sections in the chapter
33.1 Introduction
33.2 Some basic formulae
33.3 Domain and range of trigonometrical functions
33.4 Sum and difference formulae
33.5 Sum and difference into products
33.6 Product into sum or difference
33.7 T-ratios of the sum of three or more angles
33.8 Values of trigonometrical ratios some important angles and some important results.
33.9 Expressions of sin A/2 and cos A/2 in terms of sin A.
33.10 Maximum and minimum values of trigonometrical functions
Study Plan
Day 1
33.1 to 33.4
33.1 Introduction
33.2 Some basic formulae
33.3 Doman and range of trigonometrical functions
33.4 Sum and difference formulae
Do objective type exercises 58,
Day 2
33.5 Sun and difference into products
33.6 product into sum and difference
Do objective type exercises 84,91, 92, 104, 115,
Day 3
33.7 T-ratios of the sum of three or more angles
Do objective type exercises 39, 44, 106, 113, 117,
Day 4
33.8 Values of trigonometric ratios of some important angles and some important results
Do objective type exercises 1,2,3,5,6,7,8,9,10,11
Day 5
33.9 Expressions of sin A/2 and cos A/2 in terms of sin A
Do objective type exercises 12,13,14,15,16,17,19,20,21,22
Day 6
33.10 maximum and minimum values of trigonometrical functions
O.T.E.: 4,18,23-30
Day 7
O.T.E.: 31 to 50
Day 8
O.T.E.: 51 to 70
Day 9
O.T.E.: 71 to 90
Day 10
O.T.E.: 91 to 110
Revision Period
Day 11
O.T.E.: 111 to 120
Day 12
O.T.E.: 121 to 130
Day 13
O.T.E.: 131 to 140
Day 14
O.T.E.: 141 to 148
Day 15
Fill in the blanks type exercise: 1 to 10
Day 16
Fill in the blanks type exercise: 11 to 20
Day 17
Fill in the blanks type exercise: 21 to 30
Day 18
Fill in the blanks type exercise: 31 to 36
Day 19
True/false type exercise: 1 to 12
Day 20
Fill in the blanks type exercise: 13 to 25
Day 21
Practice Exercise: 1 to 10
Day 22
Practice Exercise: 11 to 20
Day 23
Practice Exercise: 21 to 32
Day 24
Formula Revision
Day 25
Formula Revision
Revision facilitator
33.1 Introduction
33.2 Some basic formulae
33.3 Domain and range of trigonometrical functions
33.4 Sum and difference formulae
33.5 Sum and difference into products
33.6 Product into sum or difference
33.7 T-ratios of the sum of three or more angles
33.8 Values of trigonometrical ratios some important angles and some important results.
33.9 Expressions of sin A/2 and cos A/2 in terms of sin A.
33.10 Maximum and minimum values of trigonometrical functions
33.1 Introduction
33.2 Some basic formulae
33.3 Domain and range of trigonometrical functions
33.4 Sum and difference formulae
33.5 Sum and difference into products
33.6 Product into sum or difference
33.7 T-ratios of the sum of three or more angles
33.8 Values of trigonometrical ratios some important angles and some important results.
33.9 Expressions of sin A/2 and cos A/2 in terms of sin A.
33.10 Maximum and minimum values of trigonometrical functions
Study Plan
Day 1
33.1 to 33.4
33.1 Introduction
33.2 Some basic formulae
33.3 Doman and range of trigonometrical functions
33.4 Sum and difference formulae
Do objective type exercises 58,
Day 2
33.5 Sun and difference into products
33.6 product into sum and difference
Do objective type exercises 84,91, 92, 104, 115,
Day 3
33.7 T-ratios of the sum of three or more angles
Do objective type exercises 39, 44, 106, 113, 117,
Day 4
33.8 Values of trigonometric ratios of some important angles and some important results
Do objective type exercises 1,2,3,5,6,7,8,9,10,11
Day 5
33.9 Expressions of sin A/2 and cos A/2 in terms of sin A
Do objective type exercises 12,13,14,15,16,17,19,20,21,22
Day 6
33.10 maximum and minimum values of trigonometrical functions
O.T.E.: 4,18,23-30
Day 7
O.T.E.: 31 to 50
Day 8
O.T.E.: 51 to 70
Day 9
O.T.E.: 71 to 90
Day 10
O.T.E.: 91 to 110
Revision Period
Day 11
O.T.E.: 111 to 120
Day 12
O.T.E.: 121 to 130
Day 13
O.T.E.: 131 to 140
Day 14
O.T.E.: 141 to 148
Day 15
Fill in the blanks type exercise: 1 to 10
Day 16
Fill in the blanks type exercise: 11 to 20
Day 17
Fill in the blanks type exercise: 21 to 30
Day 18
Fill in the blanks type exercise: 31 to 36
Day 19
True/false type exercise: 1 to 12
Day 20
Fill in the blanks type exercise: 13 to 25
Day 21
Practice Exercise: 1 to 10
Day 22
Practice Exercise: 11 to 20
Day 23
Practice Exercise: 21 to 32
Day 24
Formula Revision
Day 25
Formula Revision
Revision facilitator
33.1 Introduction
33.2 Some basic formulae
33.3 Domain and range of trigonometrical functions
33.4 Sum and difference formulae
33.5 Sum and difference into products
33.6 Product into sum or difference
33.7 T-ratios of the sum of three or more angles
33.8 Values of trigonometrical ratios some important angles and some important results.
33.9 Expressions of sin A/2 and cos A/2 in terms of sin A.
33.10 Maximum and minimum values of trigonometrical functions
Monday, January 16, 2012
Tuesday, May 20, 2008
Revision Material Ch 33. Trigonometric Ratios - 1
I.
Function-------------Domain--------------Range
Sin θ------------------R----------------[-1,1]
cos θ------------------R-----------------[-1,1]
tan θ------------R-{(2n+1)π/2, nЄI}------R=(-∞. ∞)
II.Basic relations
sin θ*cosec θ = 1
cos θ*sec θ = 1
tan θ*cot θ = 1
sin² θ + cos² θ = 1
sec²θ - tan²θ =1; 1 + tan² θ = sec² θ
cosec²θ - cot²θ = 1; 1+cot2θ = cosec²θ
III. Allied or Related angles
1. sin (-θ) = -sin θ
2. cos(-θ) = cos θ
IV.Compound angles
1. sin (A+B) = sin A cos B + cosA sin B
2. sin 2A = 2 sin A cos A
3. sin (A-B) = sin A cos B - cos A sin B
4. cos (A+B) = cos A cos B -sin A sin B
5 cos 2A = cos²A - sin²A
6. cos (A-B) = cos A cos B + sin A sin B
Function-------------Domain--------------Range
Sin θ------------------R----------------[-1,1]
cos θ------------------R-----------------[-1,1]
tan θ------------R-{(2n+1)π/2, nЄI}------R=(-∞. ∞)
II.Basic relations
sin θ*cosec θ = 1
cos θ*sec θ = 1
tan θ*cot θ = 1
sin² θ + cos² θ = 1
sec²θ - tan²θ =1; 1 + tan² θ = sec² θ
cosec²θ - cot²θ = 1; 1+cot2θ = cosec²θ
III. Allied or Related angles
1. sin (-θ) = -sin θ
2. cos(-θ) = cos θ
IV.Compound angles
1. sin (A+B) = sin A cos B + cosA sin B
2. sin 2A = 2 sin A cos A
3. sin (A-B) = sin A cos B - cos A sin B
4. cos (A+B) = cos A cos B -sin A sin B
5 cos 2A = cos²A - sin²A
6. cos (A-B) = cos A cos B + sin A sin B
Revision Material Ch 33. Trigonometric Ratios - 2
sin (A+B+C) = sin A cos B cos C + sin B cos A cos C + sin C cos A cos B - sin A sin B sin C
sin 3A = 3 sin A - 4 sin³A
Cos (A+B+C) = cos A cos B cos C - cos A sin B sin C - Cos B sin A sin C - cos C sin A sin B
tan (A+B+C) = [tan A + tan B + tan C - tan A tan B tan C]/[ 1- tan A tan B - tan B tan C - tan C tan A]
Transformation Formulae
sin (A+B) + sin (A-B) = 2 sin A cos B
2 sin A cos B = sin (A+B) + sin (A-B)
sin (A+B) - sin (A-B) = 2cos A sin B
2cos A sin B = sin (A+B) - sin (A-B)
cos (A+B) + cos (A-B) = 2cos A cos B
2cos A cos B = cos (A+B) + cos (A-B)
cos (A+B) - cos (A-B) = 2sin A sin B
2sin A sin B = cos (A+B) - cos (A-B)
Therefore
2 sin A cos B = sin (A+B) + sin (A-B)
2cos A sin B = sin (A+B) - sin (A-B)
2cos A cos B = cos (A+B) + cos (A-B)
2sin A sin B = cos (A+B) - cos (A-B)
The products of two sines or two cosines and one sine and one cosine can be transformed into the sum or differences of two sines or two cosines.
Trigonometric ratios of multiple angles
Trigonometric ratios of angle 2A in terms of an angle A
Sin 2A = 2sin A cos A
Sin 2 A = 2tan A/(1 + tan² A)
Cos 2A = cos² A - sin² A
Cos 2A = 2cos² A – 1
Cos 2A = 1 – 2sin² A
cos 2A = (1- tan² A)/(1+ tan² A)
tan 2A = 2tan A/(1 - tan² A)
Trigonometric ratios of angle 3A in terms of an angle A
Sin 3A = 3sin A - sin³ A
cos 3A = 4cos³ A – 3cos A
tan 3A = (3tan A - tan³ A)/(1-3tan² A)
Trigonometric ratios of sub-multiple angles
Trigonometric ratios of angle A in terms of an angle A/2
sin A = 2sin A/2 cos A/2
sin A = (2tan A/2)/(1 + tan² A/2)
cos A = cos² A/2 - sin² A/2
cos A = 2cos² A/2 – 1
cos A = 1 – 2sin² A/2
cos A = (1- tan² A/2)/(1+ tan² A/2)
tan A = (2tan A/2)/(1 - tan² A/2)
Trigonometric ratios of angle A in terms of an angle A/3
Sin A = 3sin (A/3) - sin³ (A/3)
cos A = 4cos³ (A/3) – 3cos (A/3)
tan A = (3tan (A/3) - tan³ (A/3))/(1-3tan² (A/3))
Trigonometric ratios of angle A/2 in terms of an angle cos A
cos A/2 = ±√[(1+cos A)/2]
sin A/2 = ±√[(1-cos A)/2]
tan A/2 = ±√[(1-cos A)/(1 + cos a)]
sin 3A = 3 sin A - 4 sin³A
Cos (A+B+C) = cos A cos B cos C - cos A sin B sin C - Cos B sin A sin C - cos C sin A sin B
tan (A+B+C) = [tan A + tan B + tan C - tan A tan B tan C]/[ 1- tan A tan B - tan B tan C - tan C tan A]
Transformation Formulae
sin (A+B) + sin (A-B) = 2 sin A cos B
2 sin A cos B = sin (A+B) + sin (A-B)
sin (A+B) - sin (A-B) = 2cos A sin B
2cos A sin B = sin (A+B) - sin (A-B)
cos (A+B) + cos (A-B) = 2cos A cos B
2cos A cos B = cos (A+B) + cos (A-B)
cos (A+B) - cos (A-B) = 2sin A sin B
2sin A sin B = cos (A+B) - cos (A-B)
Therefore
2 sin A cos B = sin (A+B) + sin (A-B)
2cos A sin B = sin (A+B) - sin (A-B)
2cos A cos B = cos (A+B) + cos (A-B)
2sin A sin B = cos (A+B) - cos (A-B)
The products of two sines or two cosines and one sine and one cosine can be transformed into the sum or differences of two sines or two cosines.
Trigonometric ratios of multiple angles
Trigonometric ratios of angle 2A in terms of an angle A
Sin 2A = 2sin A cos A
Sin 2 A = 2tan A/(1 + tan² A)
Cos 2A = cos² A - sin² A
Cos 2A = 2cos² A – 1
Cos 2A = 1 – 2sin² A
cos 2A = (1- tan² A)/(1+ tan² A)
tan 2A = 2tan A/(1 - tan² A)
Trigonometric ratios of angle 3A in terms of an angle A
Sin 3A = 3sin A - sin³ A
cos 3A = 4cos³ A – 3cos A
tan 3A = (3tan A - tan³ A)/(1-3tan² A)
Trigonometric ratios of sub-multiple angles
Trigonometric ratios of angle A in terms of an angle A/2
sin A = 2sin A/2 cos A/2
sin A = (2tan A/2)/(1 + tan² A/2)
cos A = cos² A/2 - sin² A/2
cos A = 2cos² A/2 – 1
cos A = 1 – 2sin² A/2
cos A = (1- tan² A/2)/(1+ tan² A/2)
tan A = (2tan A/2)/(1 - tan² A/2)
Trigonometric ratios of angle A in terms of an angle A/3
Sin A = 3sin (A/3) - sin³ (A/3)
cos A = 4cos³ (A/3) – 3cos (A/3)
tan A = (3tan (A/3) - tan³ (A/3))/(1-3tan² (A/3))
Trigonometric ratios of angle A/2 in terms of an angle cos A
cos A/2 = ±√[(1+cos A)/2]
sin A/2 = ±√[(1-cos A)/2]
tan A/2 = ±√[(1-cos A)/(1 + cos a)]
Monday, October 22, 2007
STUDY GUIDE CH.11 ELEMENTARY TRIGONOMETRY
JEE SYLLABUS
Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles,
-----------------------
Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles,
-----------------------
CH. 12. APPLICATION OF TRIGONOMETRY, TRIANGLES
JEE SYLLABUS
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle,
-------------------
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle,
-------------------
Study Guide Ch. 13. TRIGONOMETRIC EQUATIONS
JEE SYLLABUS
general solution of trigonometric equations.
-------------------------
JEE Question
Let F(x) be an indefinite integral of sin^2 x.
Statement - 1
The function F(x) satisfies F(x +Pi) = F(x) for all real x.
Because
Statement - 2
sin^2 (x+ Pi) = sin^2 x for all real x.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for statement – 1
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is Not a correct explanation for Statement – 1.
(C) Statement – 1 is True, Statement – 2 is False
(D) Statement – 1 is False, Statement – 2 is True
answer D
--------------------
JEE 2007 Paper I
The number of solutions of the pair of equations
2sin²θ - cos2θ = 0
2cos²θ - 3sinθ = 0
in the interval [0,2π] is
(A) zero
(B) one
(C) two
(D) four
Answer: C
-----------------------
general solution of trigonometric equations.
-------------------------
JEE Question
Let F(x) be an indefinite integral of sin^2 x.
Statement - 1
The function F(x) satisfies F(x +Pi) = F(x) for all real x.
Because
Statement - 2
sin^2 (x+ Pi) = sin^2 x for all real x.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for statement – 1
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is Not a correct explanation for Statement – 1.
(C) Statement – 1 is True, Statement – 2 is False
(D) Statement – 1 is False, Statement – 2 is True
answer D
--------------------
JEE 2007 Paper I
The number of solutions of the pair of equations
2sin²θ - cos2θ = 0
2cos²θ - 3sinθ = 0
in the interval [0,2π] is
(A) zero
(B) one
(C) two
(D) four
Answer: C
-----------------------
CH. 14 INVERSE OF TRIGONOMETRIC FUNCTIONS
JEE SYLLABUS
inverse trigonometric functions (principal value only).
---------------------
JEE 2007 Paper II
Let (x,y) be such that Sin‾¹(ax) + Cos‾¹(y)+Cos‾¹(bxy) = π/2
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
Column I --------------------- Column II
(A) If a = 1 and b = 0,------(p) lies on the circle
then (x, y)----------------- x² + y² = 1
===================================================
(B) If a = 1 and b = 1, ----(q) lies on
then (x, y)----------------- ( x²-1)(y²-1) =0
===================================================
(C) If a = 1 and b = 2,-----(r) lies on y = x
then (x, y)
===================================================
(D) If a = 2 and b = 2,-----(s) lies on
then (x, y)------------------ (4x²-1)(y²-1) = 0 ===================================================
Solution
(A) : (p)
(B) : (q)
(C) : (p)
(D) : (s)
-----------------------------------
inverse trigonometric functions (principal value only).
---------------------
JEE 2007 Paper II
Let (x,y) be such that Sin‾¹(ax) + Cos‾¹(y)+Cos‾¹(bxy) = π/2
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
Column I --------------------- Column II
(A) If a = 1 and b = 0,------(p) lies on the circle
then (x, y)----------------- x² + y² = 1
===================================================
(B) If a = 1 and b = 1, ----(q) lies on
then (x, y)----------------- ( x²-1)(y²-1) =0
===================================================
(C) If a = 1 and b = 2,-----(r) lies on y = x
then (x, y)
===================================================
(D) If a = 2 and b = 2,-----(s) lies on
then (x, y)------------------ (4x²-1)(y²-1) = 0 ===================================================
Solution
(A) : (p)
(B) : (q)
(C) : (p)
(D) : (s)
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