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Mathematics for imo & olympiad and academic

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Olympiad Exam Cracker Mathematics

📚Notes on Trigonometric Equations and Identities📚

A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).

In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).

Periods of various trigonometric functions are listed below:

1) sin x has period 2π

2) cos x has period 2π

3) tan x has period π

4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a

5) tan (ax+b) and cot (ax+b) have π/a as their period

6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a

7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period

➖Sum and Difference Formulae of Trigonometric Ratios

1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)

2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)

3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)

4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)

5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]

6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]

7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)

8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)

9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]

10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]

➖Double or Triple -Angle Identities

1) sin 2x = 2sin x cos x

2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1

3) tan 2x = 2 tan x / (1-tan 2x)

4) sin 3x = 3 sin x – 4 sin3x

5) cos3x = 4 cos3x – 3 cosx

6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)

➖For angles A, B and C, we have

1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC

2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC

3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C

4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]

➖List of some other trigonometric formulas:

1) 2sinAcosB = sin

Olympiad Mathematics Syllabus – 1 : Verbal and Non-Verbal Reasoning.

Section – 2 : Relations and Functions, Inverse Trigonometric Functions, Matrices and Determinants, Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations, Vector Algebra, Three Dimensional Geometry, Probability, Linear Programming.

OR

Section – 2 : Numbers, Quantification, Numerical Applications, Solutions of Simultaneous Linear Equations, Matrices, Determinants, Application of Derivatives, Integration, Application of Integrations, Differential Equations, Probability, Inferential Statistics, Index numbers, Time-based data, Financial Mathematics, Linear Programming.

Section – 3 : The syllabus of this section will be based on the syllabus of Quantitative Aptitude.

Section – 4 : Matrices, Determinants, Application of Derivatives, Integration, Application of Integrations, Differential Equations, Linear Programming, Probability

Class Olympiad mathematics Syllabus

Section – 1 : Verbal and Non-Verbal Reasoning.

Section – 2 : Sets, Relations and Functions, Principle of Mathematical Induction, Logarithms, Complex Numbers & Quadratic Equations, Linear Inequations, Sequences and Series, Trigonometry, Straight Lines, Conic Sections, Permutations and Combinations, Binomial Theorem, Statistics, Mathematical Reasoning, Limits and Derivatives, Probability, Introduction to 3-D Geometry.

OR

Section – 2 : Numbers, Quantification, Numerical Applications, Sets, Relations and Functions, Sequences and Series, Permutations and Combinations, Mathematical Reasoning, Limits, Continuity and Differentiability, Probability, Descriptive Statistics, Basics of Financial Mathematics, Straight Lines, Circles.

Section – 3 : The syllabus of this section will be based on the syllabus of Quantitative Aptitude.

Section – 4 : Sets, Relations and Functions, Sequences and Series, Permutations and Combinations, Limits and Derivatives, Straight Lines, Circles, Probability.

theorem

Pythagoras theorem is a basic relation in Euclidean geometry. It is a study of plane and solid figures and the five most important theorem under Euclidean geometry are the sum of the angles in a triangle is 180 degrees, the Bridge of Asses, the fundamental theorem of similarity, the Pythagorean theorem, and the invariance of angles subtended by a chord in a circle.

Pythagoras’ Theorem talks about, the square of the hypotenuse equals the sum of the squares of the other two sides. Look at the triangle ABC below, where BC2 = AB2 + AC2. The base is AB, the altitude (height) is AC, and the hypotenuse is BC. Thus, the formula goes like this:

a^2 + b^2 = c^2

:

A parallelogram is a 2D shape whose opposite sides are parallel to each other, It has four sides, where the pair of parallel sides are equal in length.

Perimeter of a Parallelogram = 2 (a+b)

Area of a Parallelogram = b × h.

:

A triangle has three sides and three inclusive angles. All the three angles of a triangle always add up to 180°.

Area of a Triangle = ½ × b × h

:

Cube is a solid 3D figure, which has 6 square faces, 8 vertices, and 12 edges, such that 3 edges meet at one vertex point. An example of a cube is a piece of Sugar, ice with six square sides.

Volume of a Cube = Side3 cubic units.

Lateral Surface Area of a Cube= 4 × side2 sq.units.

Total Surface Area of a Cube= 6× side2 sq. units.

exam cracker

-Adeep Yadav

:

A cuboid is a 3D figure with three sides where all the sides are not equal. All of its faces are rectangles having a total of 6 faces, 8 vertices, and 12 edges

of a Cuboid = (length+width+height) cubic units.

Surface Area of a Cuboid = 2×height (length + width) sq. units.

Surface Area of a Cuboid = 2(length × width + length × height + height × width) sq.units.

length of a Cuboid = length2 + breadth2 + height2 units.

:

A sphere is an object that is an absolutely round geometrical shape in 3D space. It is the set of all points in a space equidistant from a given point called the centre of the sphere. The distance between any point of the sphere and its centre is called the radius(R).

Volume of a Sphere = 4/3 x π x radius³ cubic units.

Surface Area of a Sphere = 4x π x radius² sq. units.

exam cracker.

:

A cone is a three-dimensional geometric shape. It formed by using a set of line segments or the lines which connect a common point, called the apex or vertex. At the base of a cone it has circular, so we can compute the value of radius. And the length of the cone from apex to any point on the circumference of the base is the slant height

Volume of a Cone = 1/3 × × π × radius² × height cubic units.

Total Surface Area of the Cone = πr(l + radius)

exam cracker

-Adeep Yadav

:

A circle is a basic 2D shape, and it is a set of points in a plane that are equidistant from the centre. The distance between the centre and any point on the circumference is called the radius.

Diameter of a Circle = 2 × Radius

Circumference of a Circle = π × Diameter or 2 × π × Radius

Area of a Circle = π × Radius2

exam cracker

-Adeep Yadav

:

A square is a 2D shape plane figure with four equal sides and all the four angles are equal to 90 degrees. Diagonals of a square are of equal length.

Area of a Square= Side2

Perimeter of a Square= 4(Side)

exam cracker

-Adeep Yadav

:

A rectangle is a 2D shape, having 4 sides and 4 corners. The rectangle is a quadrilateral with four right angles, so, each angle is 90°. The sum of all the interior angles is equal to 360 degrees. The opposite sides are parallel and equal to each other. Diagonals of a rectangle have the same length.

Perimeter of a Rectangle = 2(Length+Breadth)

Area of a Rectangle = Length×Breadth

exam cracker

-Adeep Yadav

#3-Dimensional Shapes:

In 3D object having three dimensions (such as height, width, and depth), like any object in the real world. In mathematical representation, it has three-axis (X, Y, and Z). Unlike 2D shapes, 3D shapes have more parameters to cover. 3D objects have some volume and Total Surface area that uses all the three dimensions i.e. length, width, and depth of the object.

exam cracker

-Adeep Yadav

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