Limit Trigonometry Function

Limit Trigonometry Function

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Limit Formula for Trigonometric Functions

limit trigonometri

Note: If π is followed by a trigonometric function the value π = 180 degrees.

The above formulas are important provisions or prerequisites to be used in calculating trigonometric limit values. Actually there are various kinds of trigonometric functions that often appear in limit problems.

In this discussion we will discuss how to solve the limit problem of trigonometric functions for x (or other variables) near zero. The following properties we use to solve the problem given.

limit fungsi

Examples of Problem Limit Trigonometric Functions

Trigonometric Limit Problem 1: Calculate the following limit,

rumus limit

Discussion Before we determine the exact value of the limit of the trigonometric function, we will estimate the limit value using a table. The table can easily be made at Ms. Excel.

contoh soal limit trigonometri

Based on the table above, we can estimate that the limit value of the function is 4. Next we specify the limit value using the limit properties of the trigonometric function.

limit fungsi aljabar

So, the limit of the given trigonometric function is 4.

Problem 2: Limit Trigonometry function try to specify a value,


Discussion We estimate the limit value of the function by using the excel function limit table below:

contoh soal limit

From the table above, we can estimate the limit value of the given function is 0.222 or 2/9. Next, we specify the limit value by using the limits of the trigonometric function.

limit tak terhingga

So, we get the limit value given function is 2/9. Well, it turns out it’s easy to find the limit for the trigonometric functions that we described above? So many reviews about () that we can write this time. Hopefully what we have learned in this article can be useful and add insight to all of us, especially for cases like trigonometry.

Trigonometry Identity Formula

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What is trigonometry? as we all know that what is meant by trigonometry is a branch of mathematics that studies the relationship of the sides and angles of a triangle and also the basic functions that arise because of the existence of these relations. Trigonometry is a comparison value defined in the cartesian coordinate of the elbow triangle.

Trigonometry has several functions, including sinus (sin), cosine (cos), tangent (tan), cosecan (cosec), secan (sec), and cotangen (cotan). Whereas the application of trigonometric identities in life is usually used to study science around astronomy , geography, and so on.

In general there are two or more trigonometric functions which, although they have different shapes, are graphically the same function. For example, two functions


which seems different, but both functions have graphs of trigonometric functions which can be described as follows.

grafik fungsi trigonometri

So we can conclude that although the two functions look different, but actually the two trigonometric functions are the same. This means, for every x value,

This last equation is called the trigonometric identity formula, and we will discuss it in this discussion. The following figure lists eight basic trigonometric identities.

identitas trigonometri

Note: The first three identities (in the orange color) graph of the trigonometric function are called inverse identities. The next two identities (in the green box) are called ratio identities. Meanwhile, the last three identities (in a blue box) are referred to as Pythagoras identity. The last two Pythagorean identities can be derived from the previous identity, namely cos² θ + sin² θ = 1, by dividing the two segments in a row with cos² θ and sin² θ. For example, by dividing both segments cos² θ + sin² θ = 1 with cos² θ, we get.

identitas trigonometri

Pythagoras identity

To derive the last Pythagorean identity, we must divide both cos² θ + sin² segments θ = 1 with sin² θ to get 1 + cot² θ = csc² θ.

After knowing the eight basic trigonometric identities above, we will then use those identities, together with our knowledge of algebra, to prove other identities.

Remember that trigonometric identity is a statement that contains the similarities of two forms for each variable change with the value in which the form is defined. To prove trigonometric identity, we use trigonometry substitution and algebraic manipulation with purpose.

Changing the shape on the left side of the identity into a shape like on the right side, or changing the shape on the right side of the identity to form as in the left side.

One thing that must be remembered in proving trigonometric identity is that we have to work on each segment separately. We must not use algebraic properties that involve both segments of identity — such as the sum of the two segments of the equation. Because, to do this, we must assume that the two segments are the same, which is something we will prove. In essence, we should not treat problems as an equation.

We prove trigonometric identity to build our ability to convert one form of trigonometric function into another. When we meet problems in other topics that require identity verification techniques, we usually find that the solution to the problem depends on how to change the form that contains trigonometry into a simpler form. In this case, we don’t have to always work with equations.

Ways to Prove Trigonometry Identity

  • Usually it will be easier if we manipulate more complicated equation fields first.
  • Look for a form that can be substituted with the trigonometric identity form in the trigonometric identity, so that it gets a simpler form.
  • Pay attention to algebraic operations, such as fraction addition, distributive traits, or factoring, which might simplify the paths we manipulate, or at least guide us to simplified forms.
  • If we don’t know what to do, change all trigonometric shapes to sine and cosine shapes. Maybe that can help.
  • Always pay attention to the equation fields that we do not manipulate to ensure the steps we take towards the shape in the segment.
  • In addition to the instructions above, the best way to become proficient at proving trigonometric identity is to practice a lot. The more trigonometric identities we have proven, then we will be more expert and confident in proving other trigonometric identities.

Examples of Trigonometric Identity Questions

Problem 1: Prove that sin θ cot θ = cos θ.


To prove this identity, we change the shape of the left segment to the shape of the right segment.

persamaan triginometri

In this example, we change the shape on the left side to the shape on the right side. Remember, we prove identity by changing one form into another.

Problem 2: Prove that tan x + cos x = sin x (sec x + cot x).

Discussion We can begin by applying distributive properties to the right hand side to multiply the terms in parentheses with sin x. Then we can turn the right segment into an equivalent form and load tan x and cos x.

rumus trigonometri

In this case, we change the right segment to the left side.

Before we go on to the next examples, let’s list some clues that might be useful in proving trigonometric identities. Some reviews of the Trigonometry Identity formula along with examples of trigonometric equations can be written this time. Hopefully what we have discussed in this article can be useful.

Logarithmic Formula Problem

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Logarithmic formula

Well, for those of you who aren’t familiar with logarithms, here we explain about the notion of logarithms in easily understood languages. Basically the notion of logarithms is a mathematical operation that is the inverse of the exponent or lift. Example of the logarithm of the exponent


if expressed by logarithmic notation is


With the following information:

     a = base or principal number
     b = result or range of logarithms
     c = numerus or logarithmic domain.

Note, it is important for you to know before we discuss further about the logarithmic formula that writing




Logarithmic properties

Here is an example of the logarithmic properties that we will write in the logarithmic table below.

If a> 0, a ≠ 1, m ≠ 1, b> 0 and c> 0, then apply:

In essence, the nature of the formula that we need to memorize is as follows. Some basic formulas or logistical properties that we need to know:


Examples of Logarithmic Problems

1). If log 2 = a

then log 5 is …


log 5 = log (10/2) = log 10 – log 2 = 1 – a (because log 2 = a)

2). √15 + √60 – √27 = …


√15 + √60 – √27

= √15 + √ (4 × 15) – √ (9 × 3)

= √15 + 2√15 – 3√3

= 3√15 – 3√3

= 3 (√15 – √3)

3). log 9 per log 27 = …


log 9 / log 27

= log 3² / log 3³

= (2. log 3) / (3. log 3) <- remember the nature of log a ^ n = n. log a

= 2/3

4). √5-3 per √5 +3 = …


(√5 – 3) / (√5 + 3)

= (√5 – 3) / (√5 + 3) x (√5 – 3) / (√5 – 3) <- times the root of the friend

= (√5 – 3) ² / (5 – 9)

= -1/4 (5-6√5 + 9)

= -1/4 (14 – 6√5)

= -7/2 + 3 / 2√5

= (3√5 – 7) / 2

5). If a log 3 = -0.3 show that a = 1/81 3√9


ª log 3 = -0.3

log 3 / log a = -0.3

log a = – (10/3) log 3

log a = log [3 ^ (- 10/3)]

a = 3 ^ (- 10/3) = 3 ^ (- 4) (3²) ^ (⅓)

a = 1/81 3√9

6). log (3a – √2) on a 1/2 base. Determine the value of a!


[log (3a – √2)] / log (0.5) = -0.5

log (3a – √2) = -0.5 log 0.5 = log (1 / √½)

3a – √2 = 1 / √½

a = (2/3) √2

Geometry Series Formula

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Understanding Geometry Series

Definition of sequences Geometry is a sequence which each tribe is obtained from the result of multiplying the previous term with a certain constant. A geometry series is a sequence that satisfies the properties of a result for a term with a previous sequence of values of constants.

For example, the geometry sequence is a, b, and c, then c / b = b / a is equal to a constant. The results for the adjacent tribe are called ratio (r).

For example a geometry series is found like the following:

U1, U2, U3, …, Un-1, Un
Then U2 / U1, U3 / U2, …, Un / Un-1 = r (constant or ratio)
Then how to determine the nth term of the geometry sequence? See the following explanation:

U3 / U2 = r then U3 = U2.r = a.r.r = ar2
Un / Un-1 = r then Un = Un-1. r = arn-2.r = arn-2 + 1 = arn-1
so it can be concluded that the geometry of the geometry n rows is Un = arn-1

a = initial rate r ratio.

Geometry Series Formula

the sum of the first n terms of a geometry sequence is called a geometric sequence. If the nth term of the geometry sequence is formulated: an = a1rn – 1, then the geometry series can be written as,

If we multiply the series with -r then add it to the original series, we get

So we get Sn – rSn = a1 – a1rn. By solving this equation for Sn, we get

The result above is the formula for the number of first n terms of the infinite geometry sequence.

Number n First Tribe Geometry
Given a geometric sequence with the first term a1 and the ratio r, the number of n the first term is

Or it can be said: The sum of the geometric sequences equals the difference from the first term and the term n + 1, then divided by 1 minus the ratio.

Example of Geometry Series Problems

Problem: Calculate the number of the first 9 terms of the sequence an = 3n .


The number of the first 9 terms can also be denoted in the sigma notation as follows.

From the series we can get the first term a1 = 3 , the ratio r = 3, and the number of terms n = 9. Using the formula for the number n first term, we get

So, the number of the first nine terms of the sequence an = 3n is 29,523.

Well, it’s easy not to calculate geometric series and infinite geometry sequences above? We feel that the discussion of geometric series formulas along with examples of geometry sequences and the answers to their discussion can be written this time. Hopefully what we have learned in this article can be useful especially for those of you who are studying math material.

Quadratic Equation Formula

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General Form of Quadratic Equations

is a form of mathematical model that states the relationship is equal to (=) and the highest rank is two. The quadratic equation has a general form:

ax2+bx+c=0 with a,b and c ϵ R and a≠0

example : 3x2+4x+1

Determining the Roots of a Quadratic Equation

Complete the quadratic equation, namely by looking for the value of x, there are several ways that can be done such as factoring, perfecting, and with the abc formula


Factoring the type ax2 +bx=0 can be done by separating x according to the distributive properties, namely:
x (ax + b) = 0
So x = 0 and ax + b = 0 Example:
Complete 3x2 +x=0
3x2 +x=0
x (3x + 1) = 0
x = 0 or
3x + 1 = 0
3x = -1
x = -1/3
so the solution is {-1/3, 0}

Factoring ax2+bx+c=0
For quadratic equations the type ax2 + bx + c = 0 can be factored in the form of:

with p and q integers or

It can be concluded#

b = p + q
c = pq / a or ac = pq

Factoring specifies a set of resolutions from 3x2-7x-6=0

With value a = 3, b = -7, c = -6
p + q = -7, p.q = 3. -6 = -18

to determine the value of p and q which when summed yielded -7 and when multiplying yielded -18 then got p = 2 and q = -9 so that:

(3x+2)(x-(9/3) )
x= -2/3

or x-3=0

so, the set of solutions {-2/3, 3}

Perfecting Quadratic

In completing the squares of the axillary equation ax2+bx+c=0 can be done with several stages:

1.     Move the constants to the right-hand segment
2.    If a ≠ 1, for both segments with a.

3. Add it to the right side and the left side quadratic of 1/2 times the coefficient x.

x2+ b/ax+ (b/2a)^2= c/a + (b/2a)^2

4. State it in the form of a perfect quadratic on the left side.

(x+b/2a)^2= c/a + (b/2a)^2

5. Then pull the root on the right side

x+b/2a=±√(c/a + (b/2a)^2 )


Messrs solution set for x2-6+2=0


x2-6+2=0 x2 - 6x= -2 x2 - 6x+ (-6/2)^2= -2 + (-6/2)^2 x2 - 6x+9=-2+9 (x-3)2=7 x-3=± √7 x=3± √7 x1 = 3+ √7 dan x2 = 3- √7

so the solution is {3- √7, 3 + √7}

Using the Abc Formula

Looking for the root values of the quadratic equation ax2+bx+c=0 is to use the following formula:

or can be written

so the roots are:


Determine the set of solutions for the 2x2 -5x -6=0 equation using the ABC formula


2x2 -5x -6=0

With the value a = 2, b = -5, c = -6 then

Determining the Root Types of Quadratic Equations with Discriminant.

In the abc formula obtained the formula:

in the formula above there is b2-4ac called discriminant (D). By using discriminan ( D= b2-4ac ), we can determine the root types from the quadratic equation, namely:

  1. If D> 0 then the quadratic equation ax2+bx+c=0 has 2 different real roots.
  2. If D <0 then the quadratic equation ax2+bx+c=0 has no real roots.
  3. If D = 0 then the quadratic equation ax2+bx+c=0 has the same 2 real roots.


Determine the type of root quadratic equation 3x2 -5x +2=0 , without first determining the roots.


With the values a = 3, b = -5 and c = 2 then
D = (-5)2-4.3.2
D = 1

Because D> 0 then the quadratic equation 3x2 -5x +2=0 has 2 different real roots.

Amounts and Results of the Roots of the Quadratic Equation

If you have obtained the roots of the quadratic equation, you can dance the results of the times and the sum of the roots of the quadratic equation. But there is an event where you are instructed to look for the results of the times or the number of root roots of the quadratic equation without getting the roots first.

Suppose the quadratic equation ax2+bx+c=0 has roots x1, x2 :

So, the formula for adding the roots of quadratic equations is:

x1 + x2 = -b / a

while for the root-root multiplication formula the quadratic equation is

So the formula for multiplying the roots of the quadratic equation is

x1.x2 = c / a

Symmetry forms of roots of quadratic equations

  1.     x12+ x22 =(x1+x2)2-2 x1x2
  2.     x13+ x23 =(x1+x2)3-3 x1x2(x1+x2)
  3.     x14+ x24 =( x12+ x22)3– 2(x1x2 )2


Foreheaded x1, x2 is the roots of the quadratic equation x2-3x+5=0 , specify the value of:

  1.      x1 + x2
  2.      x1x2
  3.    x12+ x22
  4.      (x1 / x2) + (x2 / x1)



a = 1, b = -3, and c = 5

menyusun persamaan kuadrat baru

Arrange new quadratic equations

Arrange equations when the roots are known

If a quadratic equation has the roots of x1 and x2, the quadratic equation can be expressed in the form of:

(x- x1) (x- x2) = 0


Known quadratic equations whose root roots are -2 and 3.


x1 = -2 and x2 = 3
(x – (- 2)) (x-3) = 0
(x + 2) (x + 3)

Arrange quadratic equations if the number and yield times of the roots are known.

If it is known that a quadratic equation has roots x1 and x2 and is known (x1 + x2) and (x1.x2), the quadratic equation can be formed as follows:

x2-( x1+ x2)x+(x1.x2)=0


Determine the quadratic equation whose roots are 3 and -1/2


x1 = 3 and x2 = -1/2

x1 + x2 = 3 -1/2 = 6/2 – 1/2 = 5/2

x1.x2 = 3 (-1/2) = -3/2

then the quadratic equation is

x2-( x1+ x2)x+(x1.x2)=0

x2– 5/2 x – 3/2=0 each segment multiplied by 2


That was the material friend from us about quadratic equations.

Prime and Composite Numbers

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1. Prime Numbers

Prime numbers are positive integers that are more than one. Prime numbers have only two numbers dividing the numbers themselves and 1. The first ten prime numbers are 2,3,5,7,11,13,17,19,23 and 29.

        Examples: 2 and 3 are prime numbers.

4 is not a prime number because 4 is divided

Definition: An original parameter P, P> 1 is called a prime number if and only if P has a dividing factor of 1 and P itself.

2. Composite Numbers

A composite number is a positive integer if the number has another divisor except the number itself from 1. The composite number has more than 2 set dividers. The first ten composite numbers are 4,6,8,9,10,12,14,15 and 18.

        Example: 2×2 = 4 or 2x2x2 = 8

where there are multiples of 2 or more prime numbers.

Definition: An natural number q, called a composite number, if and only if q has more than two factors, namely factor 1 and itself.

Forms Completion of Algebra

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Who ever heard the word algebra? This is a branch of mathematics in problem solving using letters to represent numbers. Derived from Arabic, al-jabr which means completion. Do you know who the inventor is? He is a scholar named Al-Khawarizmi. Now, let’s look more closely at definitions and forms of algebra in more depth!

Algebraic uses are used for many fields of study such as mathematics, chemistry, biology, economics, and so on. Well usually, before being resolved, problems are written first in the form of algebra. How is that?

A. Form of Algebra

Consists of constants (fixed values) and variables (changing values) through addition, subtraction, multiplication, division, rebar and rooting operations. Example:

You can understand the examples above if you know the definitions of terms, factors, coefficients, constants, similar and non-similar tribal variables.

bentuk-bentuk aljabar

1. Tribe is a part of an algebraic form separated by a sign – or +

     9a + 2b consists of two terms namely 9a and 2b.
     3n2 – 2n – n consists of three terms, namely 3n2, 2n, and n.

The mention of two tribes is called binom, three tribes are called trinom, while many tribes are named polynomials. However, if only a tribe is usually called a single tribe.

2. Factors are numbers that divide up other numbers or times

For example: m x n x o or m.n.o or mno. Then, the factors are m, n, and o.

3. The coefficient is a number factor in a product with a variable.

If there is a coefficient whose value is equal to 1, then you don’t need to write it. If 1a – 1b – 1c simply write a – b – c.

Example: 5×3 + 2y – 2 then 5 is the coefficient of x3, while 2 is the coefficient of y.

4. Constants are symbols that state a certain number (constant / fixed number).

Example: 9a2 + 8b – 3 then term 3 is a constant

5. Similar and non-similar tribes

It is said to be similar if it contains variables and ranks of the same variable. If both are different, it is called a non-similar tribe.

Example: 2pq + 5pq then called a similar tribe, while 2xy + 3n is called a non-similar tribe.

B. Calculating operations

You notice the component below!

     5 + 5 + 5 abbreviated to 3 x 5 or 3 (5)
     n + n is abbreviated to 2 x n or 2n
     4 x 3 x a x b is abbreviated as 12ab

Understood? Now we try to enter the number in the count operation.

1. Addition and repetition of algebraic forms

Problems example:

     Simplify the form of 5a – 2b + 6a + 4b – 3c

5a – 2b + 6a + 4b – 3c = 5a + 6a – 2b + 4b – 3c

= (5 + 6) a + (-2 + 4) b – 3c

= 11a + 2b – 3c

     Subtract 9a – 3 from 13a + 7

(13a + 7) – (9a – 3) = 13a + 7 – 9a + 3

= 13a – 9a + 7 + 3

= 4a + 10

2. Addition and subtraction of algebraic forms according to lanes or similar tribal columns

bentuk-bentuk aljabar

3. Declare multiplication of constants with two terms as the sum or difference

Distributive multiplication of addition and subtraction

Addition: a x (b + c) = ab + ac

Reduction: a x (b – c) = ab – ac

By using distributive multiplication, then the multiplication of constants with two terms can be expressed as the amount or difference. Example:

bentuk-bentuk aljabar

Start Studying Math Calculus

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In general, calculus material is a branch of mathematics that studies about problems of change. The essence of the basic calculus concept is the change in numbers used in mathematical calculations. There are several major lessons learned in this topic, namely limit function, differential (derivative), integral, and area & volume of rotating objects.

The word ‘calculus’ is taken from the Latin calculus which means small stone. This is because previous people still used small stones to do mathematical calculations. This field was first developed by 2 great scientists, Sir Isaac Newton and Gottfried Leibniz. Newton developed differential calculus, while Leibniz developed integral calculus.

This material is very important material in various sciences, especially mathematics. For mathematics, this material can be a way out for you when you cannot solve a mathematical problem using an algebraic formula.

Broadly speaking, examples of calculus problems are material that are very important in various sciences, including mathematics. The advantages of solving mathematical problems that are difficult to solve are one of the factors why this material is widely studied and one of the important sciences in mathematics.