Graphing Trigonometric Functions Tutoring

Graphing of trigonometric functions is frequent, as we construct, itself imitate about a dense period. Graphing of trigonometric function is helpful in educating science and engineering.

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Graphing trigonometric functions is taken to modify the essential trigonometric functions they are sine, cosine, tangent, cosecant, secant and cotangent. Sine and cosine functions are continual with 2 pi and tangent and cotangent functions are continual with pi.

Tutoring acts as an important role in education through online. Tutors can teach topic wise to students and correct their doubts through online. 

Example to graphing trigonometric functions tutoring:

Tutoring for graphing trigonometric functions example problem 1:

Tutoring for trigonometric function f(x) = 6tanx, execute the graphing for the trigonometric functions.

Solution:
Step 1: Trigonometric functions f(x) = 6tan x

Step 2: In the trigonometric function insert f(x) to y, we obtain,
                          y = 6tan x

Step 3: catch the points for trigonometric functions. To catch the points we can consider the value for x,

Step 4: modify x = -7 in the given trigonometric function, we get
            y = 6tan x
            y = 6tan (-7)
            y = -5.2

Step 5: Modify x = -6 in the given trigonometric function, we get
            y = 6tan x
            y = 6tan (-6)
            y = 1.7

Step 6: Modify x = -4 in the given trigonometric function, we get
            y = 6tan x
            y = 6tan (-4)
            y = -6.9

Step 7: Modify x = -3 in the given trigonometric function, we get
            y = 6tan x
            y = 6tan (-3)
            y = 0.8

Step 8: Modify x = 0 in the given trigonometric function, we get
            y = 6tan x
            y = 6tan (0)
            y = 0
Step 9: from the accomplished value, we can sketch a graphing,

X	y
-7	-5.2
-6	1.7
-4	-6.9
-3	0.8
0	0

Step 10: Illustrate a graphing by using the table.

Tutoring for graphing trigonometric functions problem 2:

Tutoring for the trigonometric function f(x) = tan 6x – y - 6, execute the graphing for the trigonometric functions.

Solution:
Step 1: Study trigonometric functions f(x) = tan 6x – y - 6

Step 2: In the given equation modify f(x) to 0, we obtain,
                          0 = tan 6x – y - 6

Step 3: Rework on the given equation as
                        y = tan 6x – 6

Step 4: catch the points for trigonometric functions. To get the points we can believe the value for x,

Step 5: Modify x = -5 in the given trigonometric function, we get
           y = tan 6x – 6
            y = tan 6(-5) - 6
            y =0.4

Step 6: Modify x = -4 in the given trigonometric function, we get
           y = tan 6x - 6
            y = tan 6(-4) -6
            y = -3.8

Step 7: Modify x = -3 in the given trigonometric function, we get
           y = tan 6x - 6
            y = tan 6(-3) - 6
            y = -4.8

Step 8: Modify x = 0 in the given trigonometric function, we get
           y = tan 6x - 6
            y = tan 6(0) - 6
            y = -6

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Step 9: from the accomplished value, we can sketch a graphing,

X	y
-5	0.4
-4	-3.8
-3	-4.8
0	-6

Step 10: Illustrate a graphing by using the table.

Samples Quadratic Equation

The quadratic equation is an expression having an order of degree 2 for variables. First term of this sample quadratic equation is in the order of two and the second term has order of degree 1 and the last term is constant. If any of the term is not present in the quadratic equation means that the term has a value 0.

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Samples of quadratic equation:

Samples of quadratic equation:
General form of quadratic equation is
ax²+bx+c.
As we saw the first degree of first term is 2, the degree of second term is 1 and third term is constant. Here the first two terms are jointly with a,b constants.

Solving a quadratic equation:
The quadratic equation can be solved by using the following formula.
x=`(-b +- sqrt(b^2 -4ac))/(2a)`
There is chance of getting two values as answers. Because of the plus,minus signs.

Example problems for samples of quadratic equation:

Example: 1
Solve the following sample quadratic equation of expression 5x²+2x+1
Solution:

Given quadratic equation is 5x²+2x+1

We know the general quadratic equation is ax²+bx+c.

 Here a=5, b=2 and c=1.

The general formula for solving a quadratic equation is,

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

 =`(-2 +- sqrt(2^2 -4.5.1))/(2.5)`

 =`(-2 +- sqrt(4-20))/10`

 =`(-2 +- sqrt (-16))/10`

We can take square root of 16 but it has a negative sign. So we have to consider it as imaginary part and insert i into the square root value.

 =`(-2 +- 4i)/10`

 = `(-1+- 2i)/5`

Answer of the above quadratic equation is `(-1+2i)/5` or `(-1 - 2i)/5`.

Example: 2

Solve the following sample quadratic equation of expression x²+2x+1.

Solution:
Given quadratic equation is

We know the general quadratic equation is ax²+bx+c.

The general formula for solving a quadratic equation is,

                      x=`(-b +- sqrt(b^2 -4ac))/(2a)`
                      = `(-2 +-sqrt(2^2-4.1.1))/(2.1)`
                     = `(-2 +- sqrt(4-4))/2`
                     =`(-2 +- sqrt(0))/2`
                     =`-2/2`

        = -1

Answer for the above quadratic equation is -1.

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Practice problems for samples quadratic equation:

Problem: 1
Solve the following sample quadratic equation of expression2x²+3x+2.
Answer: `(-3 +sqrt(7)i)/6` or `(-3 -sqrt(7)i)/6`

Problem: 2
Solve the following sample quadratic equation of expression 4x²+3x+1
Answer: `(-3 + sqrt(7)i)/8` or `(-3 - sqrt (7)i)/8`.

Sum of Finite Geometric Series

In geometric sequence is a series by a constant ratio among following expressions. For instance, the series  `(1)/(2)+(1)/(4)+(1)/(8)+...`  is geometric, since each expression except the first know how to be get by multiplying the preceding expression by  `(1)/(2)` . Geometric series are the simplest instances of infinite sequence by finite sums. Historically, geometric sequences play a significant role in the early expansion of calculus, and they maintain to be central in learn of convergence of series.

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Sum of finite geometric series:

A geometric series, also recognized as a geometric sequence, is a series of numbers where each expression after the first is establish by multiplying the preceding one by a fixed non-zero number known the common ratio. For instance, the sequence 3, 9, 27, 81, ... is a geometric series with common ratio 3. Similarly 14, 7, 3.5, 1.75, ... is a geometric series with common proportion 1/2.
           
 The sum of the expressions of a geometric series is recognized as a geometric series. Thus, the general structure of a geometric series is  a,ar,ar²,ar³... and that of a geometric series is a+ar+ar²+ar³+....  where r ≠ 0 is the common ratio and scale factor is a.

Examples for sum of finite geometric series:

Example 1:
 Find the sum of finite n terms and the sum of first 4 terms of the geometric series 2+4+8+16+...............
Solution:
Step 1: the given series is 2+4+8+16+...............
  Step 2: Here, a=2 and r=2 and n=4
Step 3: here r >1
Step 4:     Therefore
s_n = (a(1-r^n))/(1-r)` 
 n = 4

Step 5:          `s_4 = (2(1-2^4))/(1-2)`
Step 6:               `s_4 = (2(1-2^4))/(1-2)`  
Step 7:             `S_4 = 30`

Example 2:
 Find the sum of finite n terms and the sum of first 5 terms of the geometric  series 1+3+5+7+...............

Solution:
Step 1: the given series is  1+3+5+7+..........
Step 2: Here, a=1 and r=2 and n=5

Step 3: here r >1
Step 4:     Therefore
s_n = (a(1-r^n))/(1-r)` 
n = 5



Step 5:          `s_5 = (1(1-2^5))/(1-2)`
Step 6:               `s_5 = (1(1-2^5))/(1-2)`


Step 7:             `S_5 = 31`

How to Make an Octagon

Shapes play a vital part in mathematics. In geometry, the enclosed shapes are called as polygons. The polygons are classified according to its sides. Any polygon with eight sides is said be an octagon. In this article, we shall learn how to make a regular octagon. Also we shall learn how to make irregular octagon.

How to make a regular octagon:

• Step 1: Initially make a vertical line AB.

 

• Step 2: We know that the octagon has interior angle of 135°. Using protector, mark 135° on both sides of the line AB and mark it as C and D. Make the lines from A to C and B to D. These lines should be in similar length of AB.

 

• Step 3: From C take 135° and mark it as E. And from D, take 135° and mark it as F. Make the line from C to E and D to F. The length of the line must be similar to AC and BD.

 

• Step 4: From E take 135° and mark it as G. And from F, take 135° and mark it as H. Make the line from E to G and F to H. The length of the line must be similar to CE and DF.

 

• Step 5: Now draw a line from G to H.

 

   Finally the regular octagon can be obtained as shown in the figure. This is how we can make a regular octagon.

How to make irregular octagon shape:

   An octagon is a shape with 8 sides. An eight sided enclosed shape forms an irregular polygon. Now let us draw an irregular octagon.

• Step 1: Make a line and name it as AB.

 

• Step 2: Make another line from one end (A or B).

 

• Step 3: Similarly draw eight lines.

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Note: The eighth line must form an enclosed figure. The important point to remember while drawing an octagon is that it should have eight sides enclosed. The length of the side is not a matter for irregular octagon

 

   You can obtain an irregular octagon as shown in the figure. The irregular octagon can be of any form. But it should have eight sides.

Distance of a Triangle

              

                                                     

A,B and C are three non-collinear points. The figure which surronded by segments of the lines AB, BC and CA is called  triangle having A,B and C as its vertices. All the points situated on AB, BC and CA are included in this triangle. In a triangle, an exterior angle is greater than either of the interior opposite anglesThe sum of all the angles in any triangle is 180º.

 There are types of triangle

• Equilateral triangles,
• Isosceles triangles 
• Scalene triangles
• Right Triangles

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Types of Triangles:

Right Triangles:

         In a right triangle is a triangle with a 90° right angle triangle

                                    

 

Equilateral Triangles

In equilateral triangle  all three sides are in equal length also  three angles are  equal and they are  60º in each.

                                    

Isosceles Triangles

In isosceles triangle, two sides are in equal  length. The angles opposite are also equal sides

                                    

Scalene Triangles

 A scalene triangle has no equl sides of its length. Its angles are also all different in measure.

                                   

Finding distance Of a Triangle:

Example 1:

Find the distance of triangle whose points  are (-2, -3), (-4, 4), (5,8)

Solution:

to find distance of triangle where  (x₁,y1)=(-2,-3)

(_x2,y₂)=(-4,4)

                                                         

sqrt((x2-x1)^2 +(y2-y1)^2)`

   
=`sqrt(((-4-(-2))^2+4-(-3)^2))`
=`sqrt((-4+2)^2+(4+3)^2)`
=`sqrt((-2)^2+7^2)`
=`sqrt(4+49)`
length of one side  =7.28
similarly we can find another sides

Example 2:

Find the distance of triangle whose points  are (2, 3), (4, 6),(5,0)

Solution:

to find distance of triangle where  (x₁,y1)=(2,3)

(_x2,y₂)=(4,6)

sqrt((x2-x1)^2 +(y2-y1)^2)`

=`sqrt((4-2)^2+(6-3)^2))`
=`sqrt((2)^2+3^2)`
=`sqrt(4+
length of one side =`sqrt(13)`
similarly we can find another side lengths

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Example 3:

            Find the distance of triangle whose points  are (1, -3), (2, 5),(4,6)

Solution:

          to find distance of triangle where  (x₁,y1)=(1,-3)

                                                               (_x2,y₂)=(2,5)

                                                         

                    `sqrt((x2-x1)^2 +(y2-y1)^2)`

   
                   =`sqrt((2-1)^2+(5-(-4))^2))`
             
                   =`sqrt((1)^2+9^2)`
                   =`sqrt(1+81)`
                   =`sqrt(82)`

               Length of one side =9.05
Similarly we can find another two side lengths.

How to Make a Octagon

Shapes are the important theme in Geometry. In geometry, the shapes that are enclosed shapes are said to be polygons. The type of polygon is divided on the basis of its sides. Octagon is a polygon which has eight sides. There are two types of octagon. They are regular octagon and irregular octagon. In this article, we shall discuss to make a regular octagon. Also we shall discuss to make irregular octagon.

Learn how to make a regular octagon:

• Step 1: At first, make a vertical line PQ.

• Step 2: Octagon has 135degree as its interior angle. We can use protector to mark 135° on both sides of the line PQ and mark it as R and S. Make the lines from P to R and Q to S. PR and QS are the line that has the same length of PQ.

• Step 3: Mark 135degree from R and name it as T. And from S, take 135° and mark it as U. Now make the line from R to T and S to U. The length of RT and SU must be similar to the length of PQ.

• Step 4: Make 135° from T and name it as V. Make 135° from U and name it as W. Draw  line from T to V and U to W.

• Step 5: Now make a line from V to W.

Lastly the regular octagon can be attaining as shown in the figure. This is how we can make a regular octagon.

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Learn how to make irregular octagon:

An eight sided enclosed shape forms an irregular octagon. Now we shall draw an irregular octagon.

• Step 1: Make a line and name it as PQ.

• Step 2: Make another line from P.

• Step 3: Correspondingly draw eight lines.

We can attain an irregular octagon as revealed in the figure. The irregular octagon should have eight sides. But it can be drawn in any form.

Find Vertex of Parabola Calculator

Parabola is one of the conic sections whose eccentricity is equal to  1. Eccentricity of an conic section is defined as the how it is deviating from circular  shape .The parabola has three major components.this article will help you to find vertex of parabola as a calculator.
i)Vertex
ii)Focus
iii) Directrix
iv)Axis

Expalnation to find vertex of parabola calculator

Definition of parabola :

Parabola is the locus of points whose distance from fixed line, called as  directrix, and a from a fixed point , called as focus,is equal.

Vertex of parabola :-

The vertex of parabola is the point where the parabola changes its direction.It is apoint where the parabola crosses its axis.

Features of vertex of a parabola :-

i)It lies on the parabola

ii)It  lies on axis of parabola.

iii)It is apoint equidistant from focus and directrix of parabola.

Standard forms of a parabola :-

parabolas having vertex at (0,0)

i)Y² = 4aX

ii)Y² = -4aX

iii)X² = 4aY

iv)X² = -4aY

Parabolas having vertex at (h,k)

i)(Y-k)2 = 4a(X-h)

ii)(Y-k)2 = -4a(X-h)

iii)(X-k)2 = -4a(Y-h)

iv)(X-k)2 = 4a(Y-h)

Calculation of vertex of parabola :-

i) parabola is to be transformed into one of the standard forms given above  to find the  vertex of parabola.

examples to find vertex of parabola calculator

Ex:1
Given the equation of parabola Y² = 16X find the focus of parabola
solution:
Comparing with standard form of equation
Y² = 4aX
we get vertex of parabola as
(0,0)

EX 2:FInd vertex of parabola
Y² + 2y +1 = 3x-9
solution:-
transforming into standard equation
=> (y+1)² =3(x-3)
on comparing with (Y-k)² =4a(X-h)
vertex (h,k)=(3,-1)

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Ex 3: Find vetex of parabola
X²-2x+4 = 2Y-1
solution:
Converting to standard form (X-h)² =4a(Y-k)
X²-2x+4 = 2Y+1
=> (X-2)² = 2(Y+1/2)
=> (h,k)=  (2 , -1/2).