Category: Maths Notes PPT

In Mathematics, the square root of 144 is a value which when multiplied by itself gives the result 144. For example, 12 ×12= 144. Hence, we can see the square root value of 144 is 12. It is written in the radical form as 

[sqrt{144}=12]. 

There are two methods to determine the square root value of 144. The two methods to determine the square root value of 144 are Prime Factorization Method and Long Division Method. In this article, we will learn both the methods that will help the students to find the square root of any numbers in a simpler and fastest way.

According to the equation given above, we can state that the square root of natural number 144 is  12. It is a value which when multiplied by itself gives the number 144.

Hence, 12 ×12  = 144.

 

What is the Meaning of Square Root?

The square root of any natural number is a value that is represented in the form of [X=bsqrt{b}]

It implies that x is the square root of b , where b is any natural number. We can also write it as x² = b. Hence, it is concluded that the square root of any number is equal to a number which when multiplied by itself obtains back the original numbers. For Example, 5 × 5 = 25 and it can be said that the square root of a number 25 is 5.

The symbol used to represent the square root is ‘√’

The symbol of square roots is  also known as radical. The number inside the square root is known as radicand.

 

Square Root Methods

The two methods to find the square root of a given number are:

1. Prime Factorization Method

2. Long Division  Method

Prime Factorization Method

In the Prime factorization method, we generally determine the prime factors of a given number. The prime factorization method can easily be used as we have studied about prime factors in our previous classes. This method can only be used if the number given is a perfect square. A number calculated by squaring a number is considered as a perfect square. A perfect square is an integer whose square root is always an integer . For example, 9, 36, 144 etc are  perfect squares.

As we know,144 is a perfect square. 

Therefore, the prime factors of 144 = 2 × 2 × 2 × 2 × 3 × 3

If we take the square root of both the sides, we get

[sqrt{144}] = [sqrt{2times 2times 2times 2times 3times 3}]

We can see 2 pairs of 2 and 1 pair of 3 in the above given prime factors of 144.

[sqrt{144}] = 2 × 2 × 3

[sqrt{144}]= 12

Hence, the square root of 144 is 12.

Method of Long Division

We may also use the long division approach to obtain the square root of any number. This procedure is quite useful and the quickest of all for locating the root. It can be used to find the root of imperfect squares and huge numbers, something that prime factorisation cannot do. The steps for using the long division approach are outlined below.

• Take the first digit, 1, and leave the remaining two digits, 4, alone.

• The square of 1 is now 1. As a result, if we use 1 as the divisor, quotient, and dividend, the residual is 0.

• We’ll now deduct the other two numbers as dividends and add 1 to the divisor to get our next divisor, which is 1+1 =2.

• Because the last digit is 4, either the square of 2 or the number 8 can be used as the last digit.

• As a result, we’ll add 2 to 2 and multiply by 2 to get 44, as in 22 x 2.

• As a result, the final quotient is 12, which is the answer.

The steps to find the root of 144 are listed below.

Statistics is a branch of mathematics that deals with the collection, review, and analysis of data. It is known for drawing the conclusions of data with the use of quantified models. Statistical analysis is a process of collecting and evaluating data and summarizing it into mathematical form. 

 

Statistics can be defined as the study of the collection, analysis, interpretation, presentation, and organization of data. In simple words, it is a mathematical tool that is used to collect and summarize data.

 

Uncertainty and fluctuation in different fields and parameters can be determined only through statistical analysis. These uncertainties are determined by the probability that plays a very important role in statistics. 

 

What is Statistics?

In simple words statistics is the study and manipulation of given data. It deals with the analysis and computation of given numerical data. Let us take into consideration some more definitions of statistics given by different authors here:

 

The Merriam-Webster dictionary defines the term statistics as “The particular data or facts and conditions of a people within a state – especially the values that can be expressed in numbers or in any other tabular or classified way”.

 

According to Sir Arthur Lyon Bowley, statistics is defined as “Numerical statements of facts or values in any department of inquiry placed in specific relation to each other”.

 

Statistics Examples

Some real-life examples of statistics are given below:

• To find the mean of the marks obtained by each student in a class of 40 students, the average value is the statistics of the marks obtained.

• Suppose you need to find the number of employed citizens in a city. If the city has a population of 10 lakh people, we will take a sample of 1000 people. Based on this, we can prepare the data, which is the statistic.

 

Basics of Statistics

Statistics consist of the measure of central tendency and the measure of dispersion. These central tendencies are actually the mean, median, and mode and dispersions comprise variance and standard deviation. 

 

Mean is defined as the average of all the given data. Median is the central value when the given data is arranged in order. The mode determines the most frequent observations in the given data.

 

Variation can be defined as the measure of spread out of the collection of data. Standard deviation is defined as the measure of the dispersion of data from the mean and the square of the standard deviation is also equal to the variance.

 

Mathematical Statistics

Mathematical statistics is the usage of Mathematics to Statistics. The most common application of Mathematical statistics is the collection and analysis of facts about a country: its economy, and, military, population, number of employed citizens, GDP growth, etc. Mathematical techniques like mathematical analysis, linear algebra, stochastic analysis, differential equation, and measure-theoretic probability theory are used for different analytics.

 

Since probability uses statistics, Mathematical Statistics is an application of Probability theory.

 

For analyzing the data, two methods are used:

1. Descriptive Statistics: It is used to synopsize (or summarize) the data and their properties.

2. Inferential Statistics: It is used to get a conclusion from the data. 

In descriptive statistics, the data or collection of data is described in the form of a summary. And the inferential stats are used to explain the descriptive one. Both of these types are used on a large scale.

 

There is one more type of statistics, in which descriptive statistics are transitioned into inferential stats.

 

Scope of Statistics

Statistics can be used in many major fields such as psychology, geology, sociology, weather forecasting, probability, and much more. The main purpose of statistics is to learn by analysis of data, it focuses on applications, and hence, it is distinctively considered as a mathematical science.

 

Methods in Statistics

The statistical process involves collecting, summarizing, analyzing, and interpreting variable numerical data. Some methods of statistics are given below.

• Data collection

• Data summarization

• Statistical analysis

 

What is Data in Statistics?

Data can be defined as a collection of facts, such as numbers, words, measurements, observations, quantities etc.

 

Types of Data

1. Qualitative data- it is a form of descriptive data.

Quantitative data- it is in the form of numerical information.

 

Types of quantitative data

1. Discrete data- it has a fixed value that can be counted

2. Continuous data- it has no fixed value but has a range that can be measured.

 

Collecting and Summarizing Data

Data:

A collection of observations, facts about an object is known as Data. Data can be in numbers or in statement/descriptive form.

For example, 

The statement “How many legs does this table have?”. Here, the counted (or collected) value of legs is known as data.

Data Organization of the collected data is required in order to be processed. Information can be provided by processing the data.

 

Description of Data

There are various ways to describe the data:

 

Mode:

Mode is the value that occurs very often in the list. It can be said that there is no mode value if no number is repeated in the list.

 

Median:

Median is the middle value of the list. Median divides the list into two halves. 

 

Mean:

A mean is an average of all the numbers in the list. It can be calculated by adding up all the numbers and then dividing the sum by the number of values in the list.

 

Range:

The range is the difference between the larges
t and the smallest numbers.

 

Types of Statistics

Being a broad term, there exist different models of statistics:

 

Mean

A mean is an average of two or more numerals. Mean can be computed using Mathematical mean or Geometric mean. The mathematical mean shows how well the commodity performs over the period whereas the geometric mean shows the result of the investment of the same commodity over the same period.

 

Regression Analysis

It is a statistical process that determines the relationship between variables. It is the process of understanding how the value of a dependent variable changes when any of the independent variables is changed. For example, the price of the property fluctuates due to the particular industry or sector.

 

Skewness

Skewness is the measure of the distortion from the standard distribution in a set of data. A curve is said to be skewed if it is shifted to the left or to the right. If the curve is extended towards the right side, it is known as the positive skewed and if the curve is extended towards the left side, it is known as the left-skewed.

 

Kurtosis

Kurtosis is the measure of the tailedness in the frequency distribution. Data set may have heavy-tails or light-tails.

 

Variance

Variance in statistics is the measure of the data span. It is used to compare the performances of stocks over a period of time.

 

Representation of Data in Statistics

There are various ways to represent data. For example- graphs, charts and tables. The general representation of statistical data is done with the help of:

• Bar Graph

• Pie Chart

• Line Graph

• Pictograph

• Histogram

• Frequency Distribution

Bar Graph:

It is the rectangular bar representation of data. The bars can be horizontal or vertical. The length of the bar is proportional to the value that it represents. It represents data in the form of rectangular bars having length according to the values that they represent.

There are three types of bar graphs:

1. Vertical Bar Graph

2. Horizontal Bar Graph

3. Double bar Graph

 

The Representations of Vertical and Horizontal Bar Graphs are as Shown Below:

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A double bar graph is used to represent the two sets of data in the same graph.

 

A Representation of a Double Bar Graph is Shown as Below:

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Pie Chart:

It is also known as the Circle Graph as it uses sectors of the circle to represent the data. This graph is represented in the form of a circle which is divided into a various number of sectors where each sector represents a portion of the whole division.

 

Representation of the Pie chart is as shown below:

 

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Line Graph:

A line graph is represented by the straight line which connects the data points. It is represented by a series of data points called markers. Usually, a line graph is used to represent the change of the data over the period of time. 

 

A Representation of the Line Graph is as Shown Below:

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Pictograph:

It is the representation of the frequency of data using the symbols or pictures. A symbol can represent one or more numbers of data. It represents data with the help of pictures.

 

A Representation of the Pictograph is as Shown Below:

 

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Venn Diagrams:

It is the pictorial representation which contains a box along with circles. The box represents the Sample Space and the circles represent the events. There can be three types of Venn diagrams:

1. Two or more than two separate circles (When there is no common data)

2. Overlapping Circles (When some of the data is common)

3. Circle within a circle (When the outer circle is the superset of the inner circle)

 

The Representation of Each of the Venn Diagrams is as Shown Below:

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Histogram:

It consists of rectangles Whose area is proportional to the frequency of a variable and whose width is equal to the class intervals.

Frequency Distribution: 

The frequency of a value is represented by “f”. In a frequency table, specific data and values are arranged in ascending order of the given magnitude with their corresponding available frequencies.

 

Applications of Statistics

Information around the world can be determined mathematically through Statistics. There are various fields in which statistics are used:

1. Mathematics: Statistical methods like dispersion and probability are used to get more exact information.

2. Business: Various statistical tools are used to make quick decisions regarding the quality of the product, preferences of the customers, the target of the market etc. 

3. Economics: Economics is totally dependent on statistics because statistical methods are used to calculate the various aspects like employment, inflation of the country. Exports and imports can be analysed through statistics.

4. Medical: Using statistics, the effectiveness of any drug can be analysed. A drug can be prescribed only after analysing it through statistics.

5. Quality Testing: Statistics samples are used to test the quality of all the products a Company produces.

6. Astronomy: Statistical methods help scientists to measure the size, distance, etc. of the objects in the universe. 

7. Banking: Banks have several accounts to deposit customers’ money. At the same time, Banks have loan accounts as well to lend the money to the customers in order to earn more profit from it. For this purpose, a statistical approach is used to compare deposits and the requesting loans.

8. Scienc
   e: Statistical methods are used in all fields of science. 

9. Weather Forecasting: Statistical concepts are used to compare the previous weather with the current weather so as to predict the upcoming weather.

 

There are various other fields in which statistics is used.  Statistics have a number of applications in various fields in Mathematics as well as in real life. Some of the major uses of statistics are given below:

• Applied statistics, theoretical statistics, and mathematical statistics

• Machine learning and data mining

• Statistical computing

• Statistics is effectively applied to the mathematics of the arts and sciences

• Used for environmental and geographical studies

• Used in the prediction of weather

What are Trigonometry Functions? 

Trigonometry functions define the relationships among angles and sides of a right-angled triangle. The applications of such functions are wide-ranged and may be seen within the solutions of functional equations and differential equations. For instance, the sum and difference of trigonometric identities can be represented in any periodic process. 

There are 6 trigonometric functions and they are as follows.

1. sine

2. cosine

3. tangent

4. cotangent

5. secant

6. cosecant

All the functions mentioned above also have corresponding inverse trigonometric functions. 

Different Trigonometric Identities 

Before proceeding with the sum and difference of trigonometric identities, let us go through some of the important identities.

Relations Between tan, cot, sec and cosec with Respect to sin and cos

Relation Among sin and cos 

Negative Angles Identities 

It can be seen from the identities that sin, tan, cot, and cosec amount to odd functions. On the other hand, sec and cos amount to even functions. 

Sum Difference Angles Trigonometry – What are the Angle Identities?

The angle difference identities and sum identities are used to determine the function values of any of the angles concerned. To that effect, finding an accurate value of an angle may be represented as difference or sum by using the precise values of cosine, sine, and tan of angles 30°, 45°, 60°, 90°, 180°, 270°, and 360° as well as their multiples and sub-multiples. 

The following table shows the sum and difference of trigonometric identities.

Converting Product to Sum and Difference of Trigonometric Identities

For deriving the relationship between sum and difference with that of the product of trigonometric identities compound angles have to be utilized. Below are some of the important relations.

sin (A + B) = sin A cos B + cos A sin B …………………………………  (1)

sin (A – B) = sin A cos B – cos A sin B ………………………………….  (2)

cos (A + B) = cos A cos B + sin A sin B …………………………………  (3)

cos (A – B) = cos A cos B – sin A sin B ………………………………….  (4)

Therefore, for the calculation of the product formula, it may be derived –

• 2sin A cos B = sin (A + B) + sin (A – B)

• 2sin A sin B = cos (A – B) – cos (A + B)

• 2cos A sin B = sin (A + B) – sin (A + B)

• 2cosA cos B = cos (A + B) + cos (A – B)

In deriving the formulas of the products, the conversion to sum and difference of trigonometric identities can also be done. 

Few Solved Examples 

1. Value of sin 15° with Help of Difference Formula 

First step: sin (A – B) = (sin A X cos B) – (cos A X sin B)

Second step: sin (45 – 30) = (sin 45 X cos 30) – (cos 45 X sin 30)

By substituting the respective values, sin 15° comes to: [frac{sqrt{6}-sqrt{2}}{4}]

2. Value of cos 75° with Help of Sum Formula 

First step: cos (A + B) = (cos A X cos B) – (sin A X sin B)

Second step: cos (30 + 45) = (cos 30 X cos 45) – (sin 30 X sin 45)

By substituting the respective values, cos 75° comes to: [frac{sqrt{6}-sqrt{2}}{4}]

The following points should be noted while solving these sums –

For further elaboration and clarification on the topic, you may avail of ’s online classes or download the free PDFs on sums from trigonometric identity from .

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What Does Integral Mean?

To understand surface integral, it is very important to understand what an integral means. Integration and differentiation are the two sides of calculus. The two basic operations, where on one hand differentiation helps us to examine the rates of change and on the other hand integration helps us to add up infinitesimal pieces of a whole. It will be fruitless to understand integration without an example. So here is an irregular shape divided into regular rectangles of whose area integral is what we desire to calculate.

If we add up the area of all the regular rectangles, we can find the area integral of the irregular shape. If we keep making the rectangles thinner and thinner, our approximation of the area integral of the whole irregular shape would become more and more accurate. And finally when the rectangles will be infinitely small, our area integral of the shape would be perfect. This is what an integral does. It allows us to find the area, volumes, central points and displacement. Now that we know what an integration means, let us not delay in digging what a surface integration means.

Surface Integral Meaning

A surface integral is just like a line integral. In a line integral , we integrate over a path in a plane which is one dimensional and on the surface integral, we integrate over a surface which can be two dimensional or three dimensional. Therefore, we need a new kind of integral that can integrate over objects in higher dimensions and for which we need to expand the concept from a line integral to a surface integral. 

We definitely want to know how a surface integral can be calculated and what is it used for and also a surface integral example but first we would want to know how they are defined. Therefore, the surface integral of the function f(x, y, z) over the surface S will be denoted by

                                       ∫∫_sf(x,y,z)ds ……(1)

Now, dS is considered as the area of an infinitesimal piece of the surface S. in order to define the integral (1), we have to subdivide the surface S into small pieces having area ∆Si then  pick a point (xi , yi , zi) in the i-th piece, to form the Riemann sum.

                                       ∑f(xᵢ,yᵢ,zᵢ)ΔSᵢ ……(2)    

As the  subdivision of S gets finer, the corresponding sums (2) will reach a limit where it will not be dependent on the choice of the points or how the surface was subdivided. Here, the surface integral (1) is defined to be this limit. (The surface must be smooth and not infinite in extent. Also, the subdivisions must be made reasonably or else either the limit may not exist, or it may not be unique.)

Parametric Surfaces

Before we integrate over a surface and surface integral example, we must first consider the surface itself. If we recall, we will remember that in order to calculate a scalar or vector line integral over curve C, we had to first parameterize C. Therefore, to calculate a surface integral over surface S, we have to parameterize S. And to do that, we require a working concept of a parameterized surface just like we already had a concept of a parameterized curve.

A parameterized surface can be represented as:

r(u,v) = (x(u,v), y(u,v), z(u,v))       

We can see that the parameterization includes two parameters, u and v, that is because the surface is two-dimensional, and therefore we need two variables to trace out the surface. The parameters u and v vary over a region which can be called the parameter domain. The set of points in the UV-plane can be substituted into r. Each choice of u and v in the parameter domain produces a point on the surface, just like each choice of a parameter t gives a point on a parameterized curve. The whole surface is created by making possible choices of u and v over the parameter domain. Just like line integrals, surface integrals are of two kinds:

1. A surface integral of a scalar-valued function.

2. A surface integral of a vector field.

Surface Integral of a Scalar-Valued Function 

Now that we are able to parameterize surfaces and calculate their surface areas, we are ready to define surface integrals. We can start with the surface integral of a scalar-valued function. Now it is time for a surface integral example:

Consider a surface S and its function f(x, y, z)

If S is denoted by the position vector, r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k, then the surface integral of the scalar function would be:

 ∫∫_sf(x,y,z)dS =  ∫∫_D(u,v)f[x(u,v), y(u,v), z(u,v)] ⏐[frac{∂r}{∂u}] x [frac{∂r}{∂v}]⏐dudv

Where, the range of coordinates over the domain of the UV-plane are (u,v)

The cross product which is perpendicular to the surface at a point (u,v) are [frac{∂r}{∂u}] x [frac{∂r}{∂v}]

The partial derivatives are [frac{∂r}{∂u}] and [frac{∂r}{∂v}]

Therefore, the absolute value  ⏐[frac{∂r}{∂u}] x [frac{∂r}{∂v}]⏐ can be referred to as the area element.

Surface Integrals of Vector Fields

To calculate the surface integrals of vector fields, consider a vector field with surface S and function F(x,y,z). It is continuously defined by the vector position r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k.

Now let n(x,y,z) be a normal vector unit to the surface S at the point (x,y,z). Here the surface S is smooth and has a continuous function n(x,y,z). There are two possibilities such as n(x,y,z) and n(x,y,z). 

If we make a choice, the surface will be oriented which can either be inward or outward. 

a) When the Surface S is Oriented is Outward:

 ∫∫_sF(x,y,z).dS = ∫∫_sF(x,y,z).ndS =  ∫∫_D(u,v)F[x(u,v), y(u,v), z(u,v)] . ⏐[frac{∂r}{∂u}] x [frac{∂r}{∂v}]⏐dudv

b) When the Surface S is Oriented is Inward:

 ∫∫_sF(x,y,z).dS =  ∫∫_sF(x,y,z). ndS =  ∫∫_D(u,v)F[x(u,v), y(u,v), z(u,v)] . ⏐[frac{∂r}{∂u}] x [frac{∂r}{∂v}]⏐dudv

dS = ndS = vector element of the surface. s

Solved Examples

Question 1) The equations z = 12, x² + y² ≤ 25

Describe the disk having radius 5 that lie on the plane z = 12. Consider that r is the position vector field r(x, y, z) = xi + yj + zk.

Therefore, Compute,

                                            ∫∫_sr.dS.

Solution 1) Now since the disk is very much parallel to the xy plane
, the outward unit normal is k. Hence n(x, y, z) = k and so r · n = z. Thus,

∫∫_sr.dS = ∫∫_sr.ndS = ∫∫_sr.zdS = ∫∫_D12dx dy = 300π

There is an alternative that is we can solve this problem with the help of the formula for surface integrals over graphs: 

∫∫_sF.dS = ∫∫_DF(-[frac{∂g}{∂x}]i – [frac{∂g}{∂y}]j + k)dx dy.

With g(x, y) = 12 and D the disk x² + y² ≤ 25, we will get

∫∫_sr.dS = ∫∫_D(x.0 + y.0 + 12)dx dy = 12(area of D) = 300π

Question 2) Evaluate the surface integral of the vector field F = 3x²i − 2yxj + 8k over the surface S that is the graph of z = 2x − y over the rectangle [0, 2] × [0, 2].

Solution 2) We will use the formula of the surface integral over a graph z = g(x, y) :

∫∫_sF.dS =  ∫∫_D[F . (-[frac{∂g}{∂x}]i – [frac{∂g}{∂y}]j + k)dx dy.

In this case, we will get: 

∫[_{0}^{2}]∫[_{0}^{2}](3x²-2yx,8.) . (-2,1,1)dx dy∫[_{0}^{2}]∫[_{0}^{2}](-6x²,-2yx,+8)dx dy

∫[_{0}^{2}]– 2x³ – yx² + 8x 

∫[_{0}^{2}] – 4ydy = -2y² l[_{0}^{2}] = -8.

The tangent is defined in right triangle trigonometry as the ratio of the opposite side to the adjacent side (It is applicable for acute angles only because it’s only defined this way for right triangles).  To find values of the tangent function at different angles when evaluating the tangent function, we first define the reference angle created by the terminal side and the x-axis. Then we calculate the tangent of this reference angle and determine whether it is positive or negative based on which quadrant the terminal side is in. In the first and third quadrants, the tangent is positive. In the second and fourth quadrants, the tangent is negative. The slope of the terminal side is also equal to the tangent. 

Let us discuss an introduction to Trigonometry in detail before looking at the formula. Trigonometry is a branch of mathematics concerned with the application of specific functions of angles to calculations. In trigonometry, there are six functions of an angle that are widely used. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are their names and abbreviations. In relation to a right triangle, these six trigonometric functions. The sine of A, or sin A, is defined as the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) in a triangle. The other trigonometric functions are defined similarly. These functions are properties of the angle that are independent of the triangle’s size, and measured values for several angles were tabulated before computers made trigonometry tables outdated. In geometric figures, trigonometric functions are used to calculate unknown angles and distances from known or measured angles. Trigonometry has a wide range of applications, from specific fields such as oceanography, where it is used to measure the height of tides in oceans, to the backyard of our home, where it can be used to roof a building, make the roof inclined in the case of single independent bungalows, and calculate the height of the roof etc. Here, we will discuss the tan theta formula in detail. 

How to Find the Tangent?

You must first locate the hypotenuse to find the tangent. The hypotenuse is typically the right triangle’s longest side. The next task is to decide the angle. There are only two angles to choose from. You cannot choose the right angle. After you’ve chosen an angle, you will mark the sides. The side opposite to this angle will be the opposite side and the side next to the angle is the adjacent side. After labeling the sides, you can take the required ratio. Let’s discuss ratios, what is tan theta and it’s practical applications?

What is Tan Theta? 

The length of the opposite side to the length of the adjacent side of a right-angled triangle is known as the tangent function or tangent ratio of the angle between the hypotenuse and the base.

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As discussed, the tangent function is one of the three most common trigonometric functions, along with sine and cosine. The tangent of an angle in a right triangle is equal to the length of the opposite side (O) divided by the length of the adjacent side (A). It is written simply as ‘tan’ in a formula.

⇒ tan x = O/A

tan(x) is the symbol for the tangent function which is also called the tan x formula. It is one of the six trigonometric functions that are commonly used. Sine and cosine are most often associated with the tangent. In trigonometry, the tangent function is a periodic function that is very useful.

The tan formula is as follows:

⇒ Tan = Opposite/Adjacent 

What is tan theta in terms of sine and cos?

⇒ tan x = sin x/cos x

or, tan theta = sin theta/cos theta   (here, theta is an angle)

The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side, while the cosine of an angle is the ratio of the adjacent side to the hypotenuse side.

Hence, sin x = Opposite Side/Hypotenuse Side

cos x = Adjacent Side/Hypotenuse Side

Therefore, (tan formula) tan x = Opposite Side/Adjacent Side

Finding the Tangent of the Triangle

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Angles A and B are the two angles we will deal with in this triangle. To find our tangent, we must first find out the hypotenuse. Our right angle is clear to me. Can you see it? That means our hypotenuse is directly across it, and the side that measures 5 is the hypotenuse. Can you see it? Okay, as we have our hypotenuse, let’s choose an angle to work with. We’ll choose angle B. As  B is our angle, our opposite side is the side that measures 3. Our adjacent side is the one that measures 4 because it is the only side next to angle B which is not the hypotenuse. 

This means that our tangent of angle B will be the ratio of the opposite side over the adjacent side or we can write it as 3/4 which will be equal to 0.75. Similarly, If we choose angle A, our sides will change and the tangent will be 4/3 which will be equal to 1.33. 

Tangent to Find the Missing Side

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In a few problems, we will have to find the missing side of the triangle. This particularly has real-life applications like when construction companies are building on hills. In these kinds of problems, an angle is given to us. To solve these problems, we first have to locate the side that is missing. In the triangle given above, the side missed is the adjacent side to the angle given. Right?  What we have to do now is just to write the equation from the definition of the tangent. Once we have the equation, we can move ahead, use algebra to solve for the variable which will be missing. 

By multiplying both sides by x and then dividing both sides by tan 66, we’ve identified our variable. And by using the calculator to measure tan 66, we get the answer as 2.22.

Arctan – Inverse Tangent Function

There is an inverse function for any trigonometry function, for example, tan has arctan, which works in a reverse manner. These inverse functions have the same name as the originals, but with the word ‘arc’ added to the start. So, arctan is the opposite of tan. If we know the tangent of an angle and want to know the actual angle, we use the inverse function.

Large & Negative Angles

The two variable angles in a right triangle are always less than 90 degrees. However, we can find the tangent of any angle, irrespective of its height, as well as the tangent of negative angles. We can also graph the tangent function.

Tangents will be Used to Calculate the Height of a Building or a Mountain:

You can easily find the height of a building if you know the distance from where you observe it and also the angle of elevation. Similarly, you can find another side of the triangle if you know the value of one side and the angle of depression from the top of the house. All you need to know is a side and an angle of the triangle.

Conclusion

In fields like astronomy, mapmaking, surveying, and artillery range finding, trigonom
etry emerged from the need to compute angles and distances. Plane trigonometry deals with issues involving angles and distances in a single plane. Spherical trigonometry considers applications of related problems in more than one plane of three-dimensional space. 

All You Need to know About Tan theta

Students in the regular classes learn about the Triangles and various geometric measurements and operations performed over them. Trigonometry is another branch of mathematics concerned with the measurements of angles and sides corresponding to it in any right-angle triangle. Most commonly heard of functions in introductory chapters of Trigonometry are Sine theta (sin), Cosine theta (cos), tangent theta (tan), cotangent theta (cot), secant theta (sec), and cosecant theta (codec). As we all know a right angle triangle has three sides namely base, height and hypotenuse. These 6 functions of Trigonometry are nothing more than the different combinations of the three sides of the triangle in pairs concerning their proportions with each other. The functions are independent of the size or length of sides but are determined by the angle produced by the two corresponding sides.

Tangent theta or Tan theta is the ratio of the height of a right-angled triangle over the length of the base. Locating the hypotenuse of any triangle other than the right triangles sometimes becomes the difficult part of this approach. Tan theta is also used for obtaining the length of the missing side after measuring the distance between the point of observation and the origin of that side or video versa. But in this case, the value of the angle formed by the base and the line projecting to the top of the other side is required to be known before calculation.

In Mathematics, a theorem is a representation of a general concept that belongs to a broader theory. Proof is the method of demonstrating the correctness of a theorem. In this article, we will discuss all the important theorems in Mathematics Class 10.

 

The Maths theorem Class 10 includes theorems from circles, triangles, Pythagoras theorem, fundamental theorems of Arithmetic etc.

 

A triangle is a three-sided polygon with three vertices and three sides. It is one of the most fundamental geometric shapes. The Triangles Theorem Class 10 will help us to understand the properties of triangles. By proving Triangles Class 10 theorems the students’ logical thinking and reasoning skills will be improved and it will also help them to clearly understand the concepts of triangles.

 

The Class 10 Maths all Theorems PDF is available completely free on the platform. This PDF explains all theorems of triangles Class 10 proofs in a step by step manner.

 

Maths Theorems Class 10 

Here let us have a look at some of the important theorems in Mathematics Class 10.

Pythagoras Theorem

The Pythagoras theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle in mathematics.

 

According to Pythagoras theorem “The area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides”.

 

This theorem can be written as the Pythagorean equation, which relates the lengths of the sides a, b, and c.

 

[c^{2} = a^{2} + b^{2} ]

 

Where c is the hypotenuse and a and b are the other two sides of the triangle.

 

Midpoint Theorem

According to the midpoint theorem “The line segment in a triangle connecting the midpoints of two sides of the triangle is said to be parallel to its third side and is also half the length of the third side”.

 

The midpoint theorem formula is given as follows:

 

If [P(x_{1},y_{1})] and [Q(x_{2},y_{2})] are the coordinates of the two given endpoints of a line then the midpoint formula is given by

 

Midpoint [= (x_{m},y_{m})] = [frac{{(x_{1} + x_{2})}}{2}] , [frac{{(y_{1} + y_{2})}}{2}]

 

Remainder Theorem

The remainder theorem states that when a polynomial [f(x)] is divided by a linear polynomial [(x – a)], the remainder is the same as [f(a)].

 

The proof of the remainder theorem is as follows:

 

The proof for the polynomial remainder theorem is derived from the Euclidean division theorem. According to these two polynomials [P(x)] which is the dividend and [g(x)] which is the divisor, asserts the existence of a quotient [Q(x)] and a remainder [R(x)] such that

[P(x) = Q(x) times g(x) times + R(x) and R (x) = 0]

If the divisor [g(x) = x – a], where a is a constant then [R(x) = 0]

In both cases, [R(x)]  is independent of x that is [R(x)] is a constant. So we get 

[P(x) = Q(x) times (x – a) + R]

Now let us make x equal to ‘a’ in this formula, we get

[P(a) = Q(a) times (a – a) + R]

[P(x) = Q(a) times 0 +R]

[P(x) = R ]

Hence Proved.

 

Fundamental Theorem of Arithmetic

Apart from rearrangement as a product of one or more primes, the fundamental theorem of arithmetic states that any positive integer except 1 can be interpreted in exactly one way. The special factorization theorem is another name for this theorem.

 

Angle Bisector Theorem

According to the Angle Bisector Theorem, an angle bisector splits the opposite side of a triangle into two parts that are equal to the triangle’s other two sides.

 

Inscribed Angle Theorem

According to the inscribed angle theorem “An angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. As a result, when the vertex of the angle is moved around the circle, the angle does not change.

 

Ceva’s Theorem

Ceva’s theorem is an affine geometry theorem in the sense that it can be claimed and proven without using angles, regions, or lengths. As a result, triangles in an affine plane over any field are real.

 

Bayes Theorem

The Bayes theorem calculates the probability of an occurrence based on new knowledge related to that event. The formula can also be used to see how hypothetical new information affects the likelihood of an occurrence happening, assuming the new information is valid.

 

Apart from these theorems, the most important theorems of class 10 are from Triangles and Circles.

 

All Theorems of Triangles Class 10 PDF

The important triangles theorem Class 10 are as follows:

• All congruent triangles are similar, but it doesn’t mean that all similar triangles are congruent.

• If there are two triangles and if their corresponding angles are equal and also the corresponding sides are in the same ratio then the two triangles are similar triangles.

• If the sides of one triangle are proportional to the sides of the other, their corresponding angles are identical, and the two triangles are similar.

Circle Theorem Class 10

Here let us look at the important Circle Theorem Class 10.

• At the center of the circle, equal chords of a circle subtend equal angles.

• If taken from the center of the circle, the perpendicular to a chord bisects the chord.

• A circle’s equal chords are equidistant from the circle’s center.

• In a cyclic quadrilateral, the opposite angles are supplementary.

• Angles subtended by the same arc at every point on the circumference of the circle are equal to half of the angle subtended by the same arc at the center.

Conclusion

Theorems aid in the easy solution of mathematical problems, and their proofs aid in the development of a deeper understanding of the underlying concepts. Some theorems are important because they incorporate new proof methods or include a new lemma that is more useful than the proved theorem. This said the important theorems in Mathematics class 10 helps students to understand the fundamental concepts of geometry. The proof of these theorems of class 10 is created in such a way that the students will be able to grasp the concepts easily without any doubts.