Trignometric Equations

-->

An equation connecting trigonometric ratios of an angle is called a Trigonometric Identity.
An equation that gives the relation between lines and angles of a right triangle is called a Trigonometric Equation. It should be noted that an equation is satisfied for particular values of the variable whereas an identity is true for all values of the variable.

The table below gives the values of all the trigonometric ratios for different values of θ.

-->

θ T. ratios	0°	30°	45°	60°	90°
Sin θ	0	1/2	1/√2	√3/2	1
Cos θ	1	√3/2	1/√2	1/2	0
Tan θ	0	1/√3	1	√3	Not Defined
Cosec θ	Not Defined	2	√2	2/√3	1
Sec θ	1	2/√3	√2	2	Not Defined
Cot θ	Not Defined	√3	1	1/√3	0

-->

Listed below are some steps that help in determining whether an equation is Identity or not.

Step 1:- Replace the variable of the given equation by 0°, 30°, 45°, 60°, and 90° one by one.

Step 2:- If the LHS of the given equation is not equal to its RHS for some values of θ mentioned in step 1, then it is not an identity.

If all values of the variable mentioned in step 1 satisfy the given equation, then simplify the LHS and RHS of the given equation to see whether the two sides are equal or not. If the two sides are equal, the equation is an identity.

INSTALLMENT BUYING (PURCHASE) SCHEME

-->

In our day-to-day life we see lot of transactions of money like sale purchase of lands, bunglows, houses, flats etc. To make payments in transactions involving large sums of money, normally people borrow money from a bank or a finance company. The money borrowed is called LOAN.

The loan is returned in parts payable at equal intervals of time, and this called as installments.

INSTALLMENT BUYING (PURCHASE) SCHEME:- The system of purchasing as article by making payments in installments is known as installment buying scheme or installment purchase scheme. Here a customer is not required to pay a part of it at the time of purchase and rest in easy instalments, which could be monthly, quarterly, half yearly or even yearly. The time between two successive installment dates is called Payment Period.

Below are some terms which are related to Installment Buying (purchase) scheme:-

CASH PRICE: Cash price of an article is the amount which a customer has to pay as full payment of the article at the time it is purchased.

CASH DOWN PAYMENT: Cash down payment is the amount which a customer has to pay as part payment of the price of an article at the time of its purchase.

BALANCE DUE: The difference of cash price and cash down payment of an article in an installment buying scheme is known as the balance due.

AMOUNT DUE: It is the sum of the of the balance due and the interest earned on it at the end of the installment scheme.

FUTURE VALUE OF AN INSTALLMENT: Is is the sum of the value of an installment and the interest earned on it at the end of the installment scheme.

FUTURE VALUE OF ALL INSTALLMENTS: It is the sum of the future values of all the instalments at the end of the installment scheme.

Math Patterns

-->

A Pattern defines a group of numbers in which all the numbers are related with each by a specific rule. A pattern is the process of multiplying the preceding term by a constant factor. Such a sequence is called patterns. Patterns give us immense joy to find the relationship between the numbers which different number forms patterns. The constant factor is called common ratio (C.R) in patterns. We are going to explain pattern called numerical patterns.

Types of Patterns:

Below are the types of patters:

• Arithmetic Pattern.
• Geometric Pattern.

Arithmetic Pattern: Let the term a₁ is used to denote the first term, a₂ for the second term . . . and for the nth term we can use a_n and d represents the common difference between the terms. This value is equal. Then the AP becomes a₁, a₂, a_3, . . a_n. So, a₂ – a₁ = a₃ – a₂ = . . . = a_n – a_{n – 1} = d.

Then the common form of the arithmetic sequence is a, a + d, a + 2d, a + 3d, …….

An Example of Arithmetic Progression is 6, 9, 12, 15………

The n^th term of the A.P is find by the formula t_n = ar^n-1

Geometric Pattern: A geometric progression pattern is the list of terms as in an arithmetic progression but in this case the ratio of successive terms is a fixed value.

An example of a geometric progression pattern is

2,8, 32,128 ……………

r is used to denote the ratio of successive terms and a is the first term of the sequence.

The n^th term is given by t_n = ar^n-1

Here a is the first term and r is the common ratio.

Arithmetic Progression

Arithmetic Progression:

Arithmetic progression is also called as arithmetic sequence, It is a sequence that begins with an initial term a, and then each term is found by adding the common difference d.

General Form of arithmetic progression is,

a, a + d, a + 2d, a + 3d + . . .

The recursive formula is,

a_n = aⁿ⁻¹ + d.

To write the explicit form of an arithmetic series, we use

a_n = a₁ + (n − 1) d.

Example for Arithmetic Progression:

For the sequence is −2, 1, 4, 7, 10, 13, 16. . . Write the n^th term formula and find 20^th term.

Solution:

Here, the common difference is, d = 3. The n^th term formula is,

a_n = − 2 + (n − 1)3

=> a₂₅ = − 2 + (20 − 1)3

=> a₂₅ = − 2 + 19 × 3

=> a₂₅ = 55.

Laws of Probability

Following are four basic laws of probability:

Law 1 : If the probability of an event is 1, then the event must occur.
For example, the probability of each of us dying is 1. We know that dying is certain to occur.

Law 2: If the probability of an event is 0, then the event will never occur.
For example, the probability of a person who was born outside the United States becoming
its president is zero. This is the decree of the U.S. Constitution.

Law 3: The probability of any event must assume a value between 0 and 1, inclusively.
For example, the probability of its raining today is 0.7 = 70 percent. We cannot be more
than 100 percent certain that it will rain, nor we cannot be less than 0 percent certain that it
will rain.

Law 4: The sum of the probabilities of all the simple events in a sample space must be equal
to 1. Another way of saying this is to say that the probability of the sample space in any
experiment is always 1.

For example, if we consider the sample space for Example 7-4, there are 8 simple events.
By the classical approach, each simple event has an equal chance of occurring. That is,
1 A each simple event has a - chance of occurring. When we sum these probabilities, we have 8*1/8 = 1