Sine Theorem

In a triangle we can explore a number of relationships. These relation ships help us to solve a triangle. That is by knowing minimal quantities out of all the three angles and all the three sides, the remaining quantities can be figured out.

 When two sides and one angle (not the included angle) or two angles and any side are known the remaining parameters can be found. The sine theorem of triangle helps us in that.

Let us see what a sine theorem is.

The Statement of Sine Theorem-



Let ABC be any scalene type of triangle.

As per sine theorem, the following ratios are equal in any triangle.

$\frac{a}{sin A}$ = $\frac{b}{sin B}$ = $\frac{c}{sin C}$

This ratio is equal to the diameter of the circumscribing circle of the triangle.

Let us illustrate with an example.

The measures of two angles of a triangle are 30 degrees and 45 degrees. The measure of one side is 30 cm. Solve the triangle.

Let us assume A = 30o B = 45o and c = 30cm.

Using the property of sum of angles of a triangle,

C = 180o – (30o + 45o) = 105o

As per the sine theorem,

$\frac{a}{sin 30}$ = $\frac{b}{sin 45}$ = $\frac{30}{sin 105}$

Therefore,

a = 30 $\frac{sin 30}{sin 105}$ = 30 (0.52) = 15.6 cm (approximately)

b = 30 $\frac{sin 45}{sin 105}$ = 30 (0.74) = 22.2 cm (approximately)

Sine Theorem- Derivation



In the same diagram shown earlier, draw a perpendicular CD on AB and a perpendicular BE on AC.

In triangle ACD, CD = ACsin A = bsin A

In triangle BCD, CD = BCsinB = asin B

Therefore, bsinA = asinB

or,    $\frac{a}{sin A}$ = $\frac{b}{sin B}$

Similarly in triangle ABE, BE = ABsin A = csin A

and in triangle BCE, CE = BCsinC= asin C

Therefore, csinA = asinC

or,    $\frac{a}{sin A}$ = $\frac{c}{sin C}$

Combining both the results,

 $\frac{a}{sin A}$ = $\frac{b}{sin B}$ = $\frac{c}{sin C}$

Precalculus Function and Graph

Precalculus is one of the most important and interesting branch of mathematics.
Functions are basically the mappings by which elements of a given set are uniquely related to the elements of the other set. Each element on the domain have a unique image onto its co domain.
Graphs are the representation of functions on to the 2-d space . It desribes the nature of the functions . Graph can be possibly different for various classes of functions .

In this article we are going to deal with the functions and its graphs.

Precalculus Function and Graph : Examples

Example 1  : Make the graph for the precalculus function y = `e^x`

Solution : 

The `e^x` is the exponentail function . The domain is the set of all the real numbers while the range is the set of all positive real numbers.

For making the graph , we have to find the plotting points.

By putting x = 0 in the given equation , we get

y = `e^0`   =  1

By putting x = 1 in the given equation , we get

y= `e^ 1` = e = 2.71

By putting x = 0 in the given equation , we get

y = `e^2` =  7.38

x	0	1	2	3
f(x)	1	2.71	7.38	20.08

Graph is as shown :

            

Example 2  : Make the graph for the precalculus function y = ln x

Solution : 

The ln x is the natural logarithmic  function . The domain is the set of all the positive real numbers while the range is the set of all real numbers.

For making the graph , we have to find the plotting points.

By putting x = 1 in the given equation , we get

y = ln (1)   =  0

By putting x = 2 in the given equation , we get

y = ln (2)  = 0.69

By putting x = 3 in the given equation , we get

y = ln (3) = 1.09

x	0.5	1	2	3
f(x)	-0.69	0	0.69	1.09

Graph is as shown :

            

Precalculus Function and Graph : Practice Problems

Problem 1  : Make the graph for the precalculus function y = `|x|`

Problem 2  : Make the graph for the precalculus function y = sgn (x)

Standard Deviation Frequency Table

Standard deviation is an important study in Statistics. Standard deviation is the square root of the mean of the squared deviations divided by number of data. The notation of Standard deviation is σ.

The formula for calculating standard deviation is

Standard deviation(σ) =` sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

Here, n = `sum` f  number of data

In this article, we discuss about calculating standard deviation from frequency table.

Steps to Calculate Standard Deviation from Frequency Table:

Step 1: In frequency table, we calculate sum of fx2.

Step 2: Then we calculate mean bar(x) from frequency table.

Mean `bar(x)` = `(sum fx)/"n"`

Step 3: Calculate standard deviation using formula.

Let us see example problems for calculating standard deviation.

Example 1:

Calculate the standard deviation from the frequency table.

X = Weight(kg)	40	50	60	80
F = Frequency	6	8	10	16

Solution: 

The frequency table is,

Weight (x)	Frequency (f)	fx	x2	fx2
40	6	240	1600	9600
50	8	400	2500	20000
60	10	600	3600	36000
80	16	1280	6400	102400
Total	40	2520		168000

Now we are going to calculate mean bar(x) using  formula,

Mean `bar(x)` = `(sum fx)/n`

From frequency table, we know that,

`bar(x)` = `2520/40`

`bar(x)` = 63

Now we are going to calculate standard deviation from formula.

Standard deviation(σ) = `sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

From frequency table, we know that,

Standard deviation(σ) = `sqrt((168000/40)-63^2)`

Standard deviation =  `sqrt(4200-3969)`

Standard deviation = `sqrt(231)`

Standard deviation = 15.2

Therefore, standard deviation of frequency table is 15.2.

Another Example Problem for Calculating Standard Deviation from Frequency Table:

Example 2:

Calculate the standard deviation from the frequency table.

X = Marks	30	40	50	60
F = Frequency	5	7	9	15

Solution: 

The frequency table is,

 

Marks (x)	Frequency (f)	fx	x2	fx2
10	10	100	100	1000
20	11	220	400	4400
30	14	420	900	12600
40	16	640	1600	25600
Total	51	1380		43600

Now we are going to calculate mean bar(x) using  formula,

Mean `bar(x)` = `(sum fx)/n`

From frequency table, we know that,

`bar(x)` = `1380/51`

`bar(x)` = 27.1

Now we are going to calculate standard deviation from formula.

Standard deviation(σ) = `sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

From frequency table, we know that,

Standard deviation(σ) =` sqrt((43600/51)-(27.1)^2)`

Standard deviation =  `sqrt(854.9-734.4)`

Standard deviation = `sqrt(120.5)`

Standard deviation = 10.9

Therefore, standard deviation of frequency table is 10.9.

Elimination Using Addition and Subtraction

In systems of equations where the coefficients of terms contain the same variable are opposites, the elimination method can be applied by adding the equations. If the coefficients of those terms are the similar, then the elimination method can be done by subtracting the equations. Let us some example problems for elimination using addition and subtraction.

Example Problems of Elimination Using Addition and Subtraction:

Example 1:

Solve: a-b=3 and 3a+b=1

Solution:

Step 1:  Here we are going to solve these equations.

Step 2: We need to add the two equations. Here the coefficients of the b terms,(-b+b)are opposite. So we can easily cancel the terms.

Step 3: When we simplify we get 4a=4.therefore a= 1.

Now we need to calculate the value of b.

Step 6: so, we plug in a value in any equation to find b.

Step 7: here the second equation is 3a+b=1.

Step 8: So,3(1)+b=1.Therefore b = -2

How to check the solution:

We know that a and b value now. So substitute a and b value in any equation. Let us take the first equation.

Step 1: a-b=3.The value of a is 1, and the value of b is -2.

Step 2: Therefore, 1- (-2)=3 .

Step 3: When we add both the sides, we get 3=3.

Suppose if we get a different answer on both the side, the answer must be wrong.



Example 2:

Solve: u+v=3 and –u+2v=6.

Solution:

Step 1:  Here we are going to solve these equations.

Step 2: We need to add the two equations. Here the coefficients of the b terms,(-u+u)are opposite. So we can easily cancel the terms.

Step 3: When we simplify we get 3v=9.therefore v= 3.

Now we need to calculate the value of u.

Step 6: so, we plug in v value in any equation to find u.

Step 7: here the first equation is u+v=3.

Step 8: So,u + 3 = 3.Therefore u = 0.



How to check the solution:

We know that a and b value now. So substitute a and b value in any equation. Let us take the first equation.

Step 1: u+v=3 The value of u is 0, and the value of v is 3.

Step 2: Therefore,0+3=3. 3 = 3

Step 3: When we add both the sides, we get 3=3.

Suppose if we get a different answer on both the side, the answer must be wrong.

These are the example problems of elimination using addition and subtraction.

Practise Problems of Elimination Using Addition and Subtraction:

1) X+y = 7 and x-y =9

2) 2s-r =12 and s+r = -27



Answer key :

1) X = 8 and y = -1

2) S = -22 and r = -5.

Number Divisible by 4

A number divisible by 4 means nothing but a Division operation. Each and every number should divided by 4,that is called number divisible by 4.Four is a even number, When the number is even we can get the even integer numbers ,When we divided the odd number mean We cannot get the real number ,Only got the fraction numbers. Division is a one of the arithmetic operation. Arithmetic operations 1) Addition  2) Subtraction 3) Multiplication 4) Division

Step by Step Number Divisible by 4:


Form of Manual division method,

a / b = c ,where

Here number divide by 4 so we can use the constant of divisor 4

a = dividend.

b = 4 is called as divisor.

c = quotient.

Example:

12 / 4 = 3

Example problem 1: 32 divide by 4

32 divide by 4

Solution:

From the problem

32 is a dividend

4 is a divisor

In numerically it can written as 32/4

Step 1:

First we find the how many 4’s are available in the dividend

Step 2:

After find the  no of multiples in the dividend, We got the answer

32/4=8

Four is the quotient of 8

Remaining should be zero

Using Algebra Division Number Divisible by 4

Example 2: Using algebra division Number divisible by 4

Algebra division example using polynomial:

4x2+4x+4 / 4

4x2+4x+4 is dividend

4 is divisor

x2+x+1   is quotient of (4x2+4x+4) / 4

Example algebra division problem:

Division of: (42+4x+4)/4

Step 1:

(4X2+4x+4) is dividend

4 is the divisor

First we can arrange the terms

Like x2+x3+x mean we can change x3+x2+x

Step 2:

Now  we can  divide the first term of the dividend

by the1st  term of the divisor, it mean 4x2/4=x2 .It gives the first terms of quotient.

Step 3:

Now we got the first term of  quotient and then subtract the Multiplication of first terms quotient and dividend 4x2-4=x2

Step 4:

Again we can  divide the second term term of the dividend   by the first term of the divisor 4, it mean   4x/4 =1 .It should provide the second terms  after then multiply the first term with quotient after  then subtract

Step 5:

Same procedure for the constant term Now we got the final answer  x2+x+1

Quadratic Equations Activity

An equation with one variable, in which  the highest power of the variable is two is called a quadratic equation.

For example, ( i ) 3x2 + 5 x - 8 = 0

(ii) 2y2 - 48 = 0

(iii) 6x2 + 5x = 0

(iv ) y2 = 4 are all quadratic equations. Let us do some activity or problems using quadratic equations.

Activities of Quadratic Equations:

Activity 1 :

Solve the quadratic equation 2x2 - 7x = 39

2x2 - 7x - 39 - 0

`=>`              2x2  - 13x + 6x - 39 = 0  ( factorising the left hand side )

`=>`              x ( 2x - 13 ) + 3 (2 x - 13 ) = 0

`=>`             ( 2x - 13 ) ( x + 3 ) = 0

2x - 13 = 0 or x + 3 = 0

`rArr`              x  =  `13/2`   or  x  =  -3

Hence the quadratic equation is solved by factorisation method.

Activity 2 :

Find the quadratic equation whose solution set is { -2, 3 }

Since solution set is { -2, 3}

we have x = -2 or x = 3

x + 2 = 0 or x - 3 = 0

`rArr`           ( x + 2 ) ( x - 3 ) = 0

`rArr`           x2 - 3x + 2x - 6 = 0

`rArr`           x2 - x - 6 = 0 is the required quadratic equation.

Activity of Quadratic Equations(continued):

Activity 3 :

Solve the quadratic equation 5x2 - 2x - 3 = 0 using the formula.

The roots of the standard quadratic equation ax2 + bx + c = 0 where a`!=` 0, are given by the formula

x   =   `( -b stackrel(+)(-) sqrt ( b^2 - 4ac )) / ( 2a)`

Comparing 5x2 - 2x - 3 = 0 with ax2 + bx + c = 0 we get a = 5, b = -2 and c = -3.

so, x = `(2 stackrel(+)(-) sqrt((-2)^2 - 4. 5. (-3))/(2.5))`

=  `(2 stackrel( +)(-) sqrt ( 64)) / ( 10)`

= `(2 stackrel(+)(-) 8)/10`

= `(2-8)/10` = 1 and `-3/5`

Hence 1 and `(-3)/5` are the roots of the given quadratic equation.

Activity 4 :

Solve the equation 2x4 - 5x2 + 3 + 0 which is reducible to quadratic equation.

Let x2 = y

Then, 2x4 - 5x2 + 3 = 0   `rArr`   2y2 - 5y + 3 + 0

`rArr`   ( y - 1 ) ( 2y - 3 ) = 0

`rArr`   y = 1 or   y   = `3/2`

When y = 1, x2 = 1 `rArr` x = 1 or -1

When y =  `3/2`   x2 = `3/2` `rArr` x = `sqrt(3/2)`   or `-sqrt(3/2)`

Hence the fourth degree equation is solved using the quadratic equation technique.

Slope Ratio Calculator

The slope is defined as ratio of change of x axis to change of y axis. The slope intercept form is y = mx + b. Where m is slope and b is y intercept.

Slope formula is (m) =`"vertical" /"horizontal"`.

The slope form (m) is = `(y_2 - y_1)/(x_2 - x_1)` = `"rise"/"run"` .

We will learn about the slope ratio calculator example problems and practice problems are given below.

Example Problems for Slope Ratio Calculator:

Slope ratio calculator:





Example problem 1:

Find the slope ratio of a line, which contains two points A (0, 1), B (10, 2).

Solution:



The slope of a line which contains two points (x1, y1) and (x2, y2) is given by,

Here, x1 = 0, x2 = 10, y1 = 1, y2 = 2.

Slope of the line, m = `(y_2 - y_1)/(x_2 - x_1)`

= `(2 - 1)/(10 - 0)`

After simplify this, we get

= `1/(10)`

Slope of the line (m) = `1/(10)`

So, the slope ratio of a line, which contains two points A (0, 1), B (10, 2) is `1/(10)` = 0.1



Example problem 2:

Find the slope ratio of a line, which contains the two points A (-13, 10), B (2, -20).



Solution:

The slope of a line which contains two points (x1, y1) and (x2, y2) is given by,

Here, x1 = -13, x2 = 2, y1 = 10, y2 = -20.

Slope of the line, m = ` (y_2 - y_1)/(x_2 - x_1)`

=  `(-20 - 10)/(2 + 13)`

After simplify this, we get

= ` (-30)/(15)`

Slope of the line (m) = -2

So, the slope ratio of a line, which contains the two points A (-13, 10), B (2, -20) is -2

These examples problem are helpful to study of slope ratio calculator.

Practice Problems for Slope Ratio Calculator:

Practice problem 1:

Find the slope of a line, which contains two points A (9, 0), B (0, 3).

Answer: Slope (m) = -0.3333

Practice problem 2:

Find the slope of a line, which contains two points A (10, 200), B (300, 20).

Answer: Slope (m) = -0.620