Math Homework Help

Fifty minutes ago if it was four times as many minutes past 3 O’ clock, how many minutes is it to 6 O’ clock?

Got the answer?
Please feel free to put it in the comments section.

Wait for the correct answer till tomorrow.

Update
The answer was posted in the comments section.

We know Geometry is one of the most ancient branches of mathematics. The big step forward in geometry after the Greeks, was the development of a new method called Co-ordinate geometry or Analytical geometry. Modern analytical geometry is also called “Cartesian” after the name of Rene Decartes (1596-1665). But the fundamental principles and methods were already discovered by Pierre de Fermat(1601-1665). Unfortunately, Fermats treatise on the subject entitled “Ad locus planos et so lidos Isagoge“(Introduction to plane and solid loci) was published only published posthumously in 1679. So Decartes came to be regarded as the unique inventor of the analytical geometry.
In co-ordinate geometry, we enlist the services of Algebra in aid of Geometry.

There is a number which is very peculiar. This number is three times the sum of its digits. Can you find the number?
Answer
It is 27
2+7=9
3.9=27

We know the formula for finding the area of a triangle is 1/2 bh, where b is the base and h is the height of the altitude from the opposite vertex to the base.

Another important formula is given below. We can use this formula when two sides and the included angle are given. We can derive Hero’s formula from this.

The area of triangle is given by

Similarly other results can also prove.

Today I have a very interesting puzzle for you. Some of you might have heard about this before. Do you know the answer? Here is the puzzle once again.

Three friends went to a restaurant for lunch. When the bill arrived, it came exactly $30. So each person put in $10 towards the bill. The waiter took the money but the manager pointed out that the customers had been overcharged by $5. The dishonest waiter took five $1 coins from the till and kept 2 for himself and gave back the other 3 back to the customers.

Each of the friends had paid $10 minus the $1 they got back for the meal. That is each paid $9 which in total is $27 for all the 3 friends. The waiter had kept $2 making a total of $27 + $2 = $29.

So what happened to the other $1?

Answer

There is no missing dollar here. The friends spent $ 27 together. From that $25 they paid for the bill and the remaining $2 had taken by the waiter. But I have one question for you -Who told you to add $2 to $27?

In a triangle ABC,

a = b cosC + c cosB
b = c coaA + a cosC
c = a cosB + b cosA

These are known as projection formulea.

( To know the the parameters refer here. )

Proof
Let ABC be any of the triangles in the above figures

In fig (i) we have
a= BC = BD+DC……………..(a)
But BD/DA= cosB andDC/CA=cosC
==>BD=c cosB
and
DC=b cosC
Putting these values in (a)
a=c cosB+ bcosC

In fig(ii) BC=BD-CD ……………………… (b)
Here CD/CA=cos(180-C)=-cosC
==>CD=-bcosC
Also,BD=ccosB
Putting these values in (b)
a= c cosB-(-b cosC)
=c cosB+ b cosC

Other results can also be proved.

Similarly the results can also be proved for Fig(iii) also.

Example:

In any triangle ABC, prove that
(b +c )cos A+(c +a) cos B+ (a + b )cos C = a+ b+ c

Answer

Formula required: Projection Formula

In any triangle ABC, a = b cos C+c cos B

We have L.H.S=(b +c )cos A+(c +a) cos B+ (a + b )cos C

=b cos A+c cos A+c cos B+a cos B+ a cos C+ b cos C

=(b cos C+c cos B) + (a cos C+ c cos A) +(a cos B+b cos A) [ using projection formula]

=a+b+c

=R.H.S

The study of Trigonometry was first started in India. The ancient Indian mathematicians Aryabhatta (A.D 476),Bhaskara I(A.D 600), Bhaskara(A.D 1114)and Brahmagupta(A.D 598) got important results. All this knowledge first went from India to Middle East and from there to Europe. The Greeks had also started the study of trigonometry but their approach was so clumsy that when the Indian approach became known, it was immediately adopted throughout the world.
In India, the predecessor of the modern trigonometric function, known as the sine of an angle, and the introduction of the sine function represents the main contribution of the sidhantas to the history of mathematics.
Baskara I gave formula to find the values of sine function for angles more than 90 degree.
The name of Thales (A.D 600) is associated with height and distances problems. He is credited with the determination of the height of pyramid in Egypt by measuring shadows of the pyramid using similarity property.

Let f:A–>B be a function. Let y0 be an element in B. then y0 is called a value of f provided there is some element, x0 in A, such that y0 = f(x0); that is, y0 is a value of
the function f if it corresponds, with respect to the rule of f, to some x0 in the set A = Dom(f).

Example
Find the value of the following function when x=-2

Find the Domain and Range of this function.

Yesterday we have learned what is a function.
Today let’s discuss about range and domain of a function.

Answers

Try to do more problems from your text.

Today we can discuss a topic from functions.

Definitions:

Function
Let A anb B be two non empty sets. A function “f” from a set A to a set B is a rule so that to each element x in A there corresponds exactly one element y in B, under f ,then we say that f is a functin from A to B and write
f:A -> B

y is called the image of x under f and is denoted by f(x). x and y are respectively called the independent variable and the dependent variable. We also say that y is a function of x and write
y=f(x)

Examples:

1. In the family of circles, the area A of the circle is a function of radius r of the circle.
   Here radius r is the independent variable and area A is the dependent variable
2. The speed of a chemical reaction increases 2 times with the addition of every 5 milligrams of a catalyst. Here the amount of catalyst is the independent variable and speed of the chemical reaction is the dependent variable.