Topics
Mathematical Reasoning
- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Contrapositive and Converse
- Introduction of Validating Statements
- Validation by Contradiction
- Difference Between Contradiction, Converse and Contrapositive
- Consolidating the Understanding
Sets
- Sets and Their Representations
- Empty Set (Null or Void Set)
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Intrdouction of Operations on Sets
- Union of Sets
- Intersection of Sets
- Difference of Sets
- Complement of a Set
- Practical Problems on Union and Intersection of Two Sets
- Proper and Improper Subset
- Open and Close Intervals
- Disjoint Sets
- Element Count Set
Sets and Functions
Relations and Functions
- Cartesian Product of Sets
- Concept of Relation
- Concept of Functions
- Some Functions and Their Graphs
- Algebra of Real Functions
- Ordered Pairs
- Equality of Ordered Pairs
- Pictorial Diagrams
- Graph of Function
- Pictorial Representation of a Function
- Exponential Function
- Logarithmic Functions
- Brief Review of Cartesian System of Rectanglar Co-ordinates
Algebra
Trigonometric Functions
- Concept of Angle
- Introduction of Trigonometric Functions
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Trigonometric Functions of Sum and Difference of Two Angles
- Trigonometric Equations
- Trigonometric Functions
- Truth of the Identity
- Negative Function Or Trigonometric Functions of Negative Angles
- 90 Degree Plusminus X Function
- Conversion from One Measure to Another
- 180 Degree Plusminus X Function
- 2X Function
- 3X Function
- Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
- Graphs of Trigonometric Functions
- Transformation Formulae
- Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Sine and Cosine Formulae and Their Applications
Coordinate Geometry
Complex Numbers and Quadratic Equations
- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- The Modulus and the Conjugate of a Complex Number
- Argand Plane and Polar Representation
- Quadratic Equations
- Algebra of Complex Numbers - Equality
- Algebraic Properties of Complex Numbers
- Need for Complex Numbers
- Square Root of a Complex Number
Calculus
Mathematical Reasoning
Linear Inequalities
Principle of Mathematical Induction
Statistics and Probability
Permutations and Combinations
- Fundamental Principles of Counting
- Permutations
- Combination
- Introduction of Permutations and Combinations
- Permutation Formula to Rescue and Type of Permutation
- Smaller Set from Bigger Set
- Derivation of Formulae and Their Connections
- Simple Applications of Permutations and Combinations
- Factorial N (N!) Permutations and Combinations
Binomial Theorem
- Introduction of Binomial Theorem
- Binomial Theorem for Positive Integral Indices
- General and Middle Terms
- Proof of Binomial Therom by Pattern
- Proof of Binomial Therom by Combination
- Rth Term from End
- Simple Applications of Binomial Theorem
Sequence and Series
Straight Lines
- Slope of a Line
- Various Forms of the Equation of a Line
- General Equation of a Line
- Distance of a Point from a Line
- Brief Recall of Two Dimensional Geometry from Earlier Classes
- Shifting of Origin
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
Conic Sections
- Sections of a Cone
- Concept of Circle
- Introduction of Parabola
- Standard Equations of Parabola
- Latus Rectum
- Introduction of Ellipse
- Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse
- Special Cases of an Ellipse
- Eccentricity
- Standard Equations of an Ellipse
- Latus Rectum
- Introduction of Hyperbola
- Eccentricity
- Standard Equation of Hyperbola
- Latus Rectum
- Standard Equation of a Circle
Introduction to Three-dimensional Geometry
Limits and Derivatives
- Intuitive Idea of Derivatives
- Introduction of Limits
- Introduction to Calculus
- Algebra of Limits
- Limits of Polynomials and Rational Functions
- Limits of Trigonometric Functions
- Introduction of Derivatives
- Algebra of Derivative of Functions
- Derivative of Polynomials and Trigonometric Functions
- Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
- Limits of Logarithmic Functions
- Limits of Exponential Functions
- Derivative of Slope of Tangent of the Curve
- Theorem for Any Positive Integer n
- Graphical Interpretation of Derivative
- Derive Derivation of x^n
Statistics
- Measures of Dispersion
- Concept of Range
- Mean Deviation
- Introduction of Variance and Standard Deviation
- Standard Deviation
- Standard Deviation of a Discrete Frequency Distribution
- Standard Deviation of a Continuous Frequency Distribution
- Shortcut Method to Find Variance and Standard Deviation
- Introduction of Analysis of Frequency Distributions
- Comparison of Two Frequency Distributions with Same Mean
- Statistics Concept
- Central Tendency - Mean
- Central Tendency - Median
- Concept of Mode
- Measures of Dispersion - Quartile Deviation
- Standard Deviation - by Short Cut Method
Probability
- Random Experiments
- Introduction of Event
- Occurrence of an Event
- Types of Events
- Algebra of Events
- Exhaustive Events
- Mutually Exclusive Events
- Axiomatic Approach to Probability
- Probability of 'Not', 'And' and 'Or' Events
- Identities Related to Sin 2x, Cos2x, Tan 2x, Sin3x, Cos3x and Tan3x.
- Deducing the Identities
- Deducing the identities like the following:-
`tan(x+-y)=(tanx+-tany)/(1+-tanxtany)", "cot(x+-y)=(cotxcoty+-1)/(coty+-cotx)`
`sinalpha+-sinbeta=2"sin"1/2(alpha+-beta)"cos"1/2(alpha+-beta)`
`cosalpha+cosbeta=2"cos"1/2(alpha+beta)"cos"1/2(alpha-beta)`
`cosalpha-cosbeta=-2"sin"1/2(alpha+beta)"sin"1/2(alpha-beta)`
Notes
1) sin (-x)= -sin x
2) cos (–x) = cos x
3) sin (x+y)= sin x cos y+ cos x sin y
4) sin (x-y)= sin x cos y- cos x sin y
5) cos (x+y)= cos x cos y- sin x sin y
6) cos (x-y)= cos x cos y+ sin x sin y
7) cos `(π/2 - x)`= sin x
8) cos `(π/2 +x)`= -sin x
9) cos (π- x)= -cos x
10) cos (π+ x)= -cos x
11) cos (2π- x)= cos x
12) sin `(π/2 - x)`= cos x
13) sin `(π/2 + x)`= cos x
14) sin (π- x)= sin x
15) sin (π+ x)= -sin x
16) sin (2π- x)= -sin x
17) tan (x+y)= `(tan x+ tan y)/(1- tan x tan y)`
18) tan (x-y)= `(tan x- tan y)/(1+ tan x tan y)`
19) cot (x+y)= `(cot x cot y- 1)/(cot y+ cot x)`
20) cot (x-y)= `(cot x cot y+1)/cot y- cot x`
21) sin2x= 2 sin x cos x
sin2x= `(2 tan x)/(1+ tan^2x)`
22) cos2x= `cos^2 x- sin^2 x`
cos2x= `2 cos^2 x- 1`
cos2x= `1- 2 sin^2 x`
cos2x= `(1- tan^2 x)/(1+ tan^2 x)`
23) tan2x= `(2 tan x)/(1- tan^2 x)`
24) sin3x= `3 sin x- 4 sin^3 x`
25) cos3x= `4 cos^3 x- 3 cos x`
26) tan3x= `(3 tan x- tan^3 x)/(1- 3 tan^2 x)`
27) cos x+ cos y= `2cos (x+y)/2 cos (x-y)/2`
28) cos x- cos y= `-2 sin (x+y)/2 sin (x-y)/2`
29) sin x+ siny= `2 sin (x+y)/2 cos (x-y)/2`
30) sin x- sin y= `2 cos (x+y)/2 sin (x-y)/2`
31) sin (x+y)+ sin(x-y)= 2 sin x cos y
32) sin (x+y)- sin (x-y)= 2 cos x sin y
33) cos (x+y)+ cos (x-y)= 2 cos x cos y
34) cos (x+y)- cos (x-y)= 2 sin x sin y
35) 2 cos x cos y = cos (x + y) + cos (x – y)
36) –2 sin x sin y = cos (x + y) – cos (x – y)
37) 2 sin x cos y = sin (x + y) + sin (x – y)
38) 2 cos x sin y = sin (x + y) – sin (x – y)
Shaalaa.com | Trigonometric Ratios of Sum & difference of two angles
Related QuestionsVIEW ALL [178]
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
