Mathematics : Trigonometry.

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry is found all throughout geometry, as every straight-sided shape may be broken into as a collection of triangles.

Another angle is often labelled θ, and the three sides are then called:

Adjacent: adjacent (next to) the angle θ.

Opposite: opposite the angle θ.
Hypotenuse: the longest side is the Hypotenuse.

Start With Tan (x) =  Sin (x) / Cos (x).

Cot (x) opposite side of the hexagon to tan.

Cosec (x) next to Cot (x).

Sec (x) next to Cosec (x).

NOTE : All “ Co “ functions are  on the right side.

CLOCK WISE

Tan (x) = Sin (x) / Cos (x)

Sin (x) = Cos (x) / Cot (x)

Cos (x) = Cot (x) / Cosec (x)

Cot (x) = Cosec (x) / Sec (x)

Cosec (x) = Sec (x) / Tan (x)

Sec (x) = Tan (x) / Sin (x)

COUNTERCLOCK WISE

Tan (x) = Sec (x) / Cosec (x)

Sec (x) = Cosec (x) / Cot (x)

Cosec (x) = Cot (x) / Cos (x)

Cot (x) = Cos (x) / Sin (x)

Cos (x) = Sin (x) / Tan (x)

Sin (x) = Tan (x) / Sec (x)

RECIPROCAL RELATIONS

Sin (x) = 1 / Cos (x)

Cos (x) = 1 / Sec (x)

Tan (x) = 1 / Cot (x)

Cosec (x) = 1 / Sin (x)

Sec (x) = 1 / Cos (x)
Cot (x) = 1 / Tan (x)

SQUARE LAW FORMULAS

Open Links given below to view questions :

Mathematics : Trigonometry - Section A Questions.

Mathematics : Trigonometry - Section B Questions.

Mathematics : Probability

Probability theory, a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

How likely something is to happen.
Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Probability Formula.

P(A) = n(E)/n(S)

Where,

P(A) is the probability of an event “A”

n(E) is the number of favourable outcomes

n(S) is the total number of events in the sample space

NOTE : Here, the favourable outcome means that the outcome of interest.

Basic  Probability  Formulas.

Probability Range , 0 ≤ P(A) ≤ 1

Rule of Complementary Events ,  P(A’) + P(A) = 1

Examples

Example 1: What is the probability that a card taken from a standard deck, is an Ace?

Solution:  Total number of cards a standard pack contains = 52

A deck of cards contain Ace = 4 cards

So, the number of favourable outcome = 4

Now, by looking at the formula,

Probability of finding an ace from a deck is,

P(Ace) = (Number of favourable outcomes) / (Total number of favourable outcomes)

P(Ace) = 4/52

= 1/13

So we can say that the probability of getting an ace is 1/13.

Example 2: Calculate the probability of getting an odd number if a dice is rolled?

Solution: Sample space (S) = {1, 2, 3, 4, 5, 6}

Let “E” be the event of getting an odd number, E = {1, 3, 5}

So, the Probability of getting an odd number,

P(E) = (Number of outcomes favorable)/(Total number of outcomes)

P(E) = n(E)/n(S) = 3/6 = ½

Question 1 : Complete the following statements

(i) Probability of an event E + Probability of the event ‘not E’ =………………………….

(ii) The probability of an event that cannot happen is …………………………. Such an event is

Called …………………………..

(iii) The probability of an event that is certain to happen is …………………………. Such an event is called ………………………….

(iv) The sum of the probabilities of all the elementary events of an experiment is ………………………….

(v) The probability of an event is greater than or equal to ………………………….and less than or equal to ………………………….

Question 2 : Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

(iv) A baby is born. It is a boy or a girl.

Question 3 : Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

Question 4 : Which of the following cannot be the probability of an event?

(A) 2/3 (B) –1.5 (C) 15% (D) 0.7

Question 5 : If P(E) = 0.05, what is the probability of ‘not E’?

Question 6 : A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out

(i) an orange flavoured candy?

(ii) a lemon flavoured candy?

Question 7 : It is given that in a group of 3 students, the probability of 2 students not having the same  birthday  is  0.992.  What  is  the  probability  that  the  2  students  have  the  same birthday?

Question 8 : A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i)red ? (ii) not red?

Question 9 : A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be

(i)                red ?           (ii) white ?       (iii) not green?

Math : Mensuration

What is Mensuration?

Mensuration is a topic in Geometry which is a branch of mathematics. Mensuration deals with length, Perimeter, area and volume of different kinds of shape- both 2D and 3D.

 

What is a 2D Shape?

A 2D shape is a shape that is bounded by three or more straight lines or a closed circular line in a plane. These shapes have no depth or height; they have two dimensions- length and breadth and are therefore called 2D figures or shapes. For 2D shapes, we measure area (A) and perimeter (P).

What is a 3D Shape?

A 3D shape is a shape that is bounded by a number of surfaces or planes. These are also referred to as solid shapes. These shapes have height or depth unlike 2D shapes; they have three dimensions- length, breadth and height/depth and are therefore called 3D figures.  3D shapes are actually made up of a number of 2D shapes. Also, know as solid shapes, for 3D shapes we measure Volume (V), Curved Surface Area (CSA), Lateral Surface Area (LSA) and Total Surface Area (TSA).

 

Important Terms

 

Perimeter (P) : The length of the boundary of a figure is called its perimeter. In other words, it is the continuous line along the periphery of the closed figure. It is represented by the alphabet P and is measures in cm/ m.

Area (A) : The surface occupied by a given closed shape is called its area. It is represented by the alphabet A and is measured in unit square- m2/ cm2.

Volume (V) : The space that is contained in a three-dimensional shape is called its volume. In other words, it is actually the space that is enclosed in a 3D figure. It is represented by the alphabet V and is measured in cm3/ m3.

Curved Surface Area (CSA) : In solid shapes where there is a curved surface, like a sphere or cylinder, the total area of these curved surfaces is the Curved Surface Area. . The acronym for this is CSA and it is measured in m2 or cm2.

Lateral Surface Area (LSA) : The total area of all the lateral surfaces of a given figure is called its Lateral Surface Area. Lateral Surfaces are those surfaces that surround the object. The acronym for this is LSA and it is measured in m2 or cm2.

Total Surface Area (TSA) : The sum of the total area of all the surfaces in a closed shape is called its Total Surface Area. For example, in a cuboid when we add the area of all the six surfaces we get its Total Surface Area. The acronym for this is TSA and it is measured in m2 or cm2.

 TOPICS   .

·         Rectangles and Squares.

·         Parallelograms.

·         Triangles.

·         Quadrilaterals.

·         Trapeziums.

·         Circles.

·         Cubes and Cuboids

·         Spheres.

·         Cylinder.

·         Cones.

·         Frustum.

·         Miscellaneous.