Calculate Angles in a Triangle

In mathematics, the triangles are nothing but the shapes which is made by 3 lines in topic geometry. In geometry, the sides are used to calculate the angles in triangles. The classification of triangles is named as Equilateral triangle, scalene triangle, Isosceles triangle, Right triangle, obtuse triangle, and acute triangle. The formula to calculate the angles in triangles are given below.

Calculate Angles in a Triangle – Formulas:

In geometry, there are different types of formulas to calculate the values of angles in the triangles. To calculate the angles value of right angle triangles we are having certain formulas based on the sides. Let as consider x, y and z are the three sides of the right angle triangles and X is the angles of the right angle triangles. Then Pythagoras theorem is used to calculate the value of angles in right triangles.

The Right Angle Triangles are shown below:



Formulas to calculate the angles of the right triangles are given as,

Sin A = x/z

cos A = y/z

Tan A = x/y

Formula to calculate the sides of the right triangles is given as,

x2 + y2 = z2

Calculate Angles in a Triangle – Examples:

Example 1: Calculate the angles A using the given data x = 5 and y = 8?

Sol:

From the given data x = 5 and y = 8

The diagram is given as,



Formula to calculate the sides of the right triangles is given as,

x2 + y2 = z2

52 + 82 = z2

25 + 64 = z2

z2 = 89

z = 9.4

Formulas to calculate the angles of the right triangles are given as,

Sin A = x/z

Sin A = 5/9.4

Sin A = .53

A = Sin-1 (.53)

A = 32.13

Example 2: Calculate the angles A using the given data x= 8 and z = 14?

Sol:

From the given data x= 8 and z = 14

The diagram is given as,



Formulas to calculate the angles of the right triangles are given as,

Sin A = x/z

Sin A = 8/14

Sin A = .57

A = Sin-1 (.57)

A = 34.84

Calculate angles in a triangle - Practice problem:

Problem 1: Calculate the angles A using the given data x= 6 and y = 17?

Answer is given below:

z = 18.02

A = 19.26

Problem 2:Calculate the angles A using the given data x= 5 and z = 14.6?

Answer is given below:

A = 11.86

Frequency Table Values that are on a Limit

A frequency table deals with the process of data that count the number of times for each set of data, the value of count with data process in a set is called as a frequency table. It has presented in a particular value of data and the table has a count of values that shows in the raw data. This is a process which can create a frequency table in a typically raw data. Let us see about the frequency table values that are on limits.

  

Frequency Table Values that are on a Limit:

The frequency table has different types which has follows.

• Frequency table of angular:

      It is define in the rate of alteration and orientation slant in the phase of a sinusoidal waveform.

• Frequency table of grouped:

      It is use to forming a group of data information in the given values.

• Frequency table of cumulative grouped:

      A cumulative frequency grouped table is a sum of frequencies in the group table satisfying the given conditions which define the entire of all grouped frequencies in a frequency distribution.

• Frequency table of simple:

       It is a function of forming a simple data collection in the given values.

• Frequency table of cumulative:

       A cumulative frequency table is the summation of frequencies that satisfying the given conditions which describe the entire of all frequencies in a frequency distribution.

Example Problem for Frequency Table of Limit Value:

Find the following table of showing the marks assignments in a set of a 10thclass of 20 students.

2	7	5	2	7	8	2	5	9	7
4	10	6	8	3	9	5	6	4	3

To set this data of information in a frequency table.

Sol:

Step 1:

        To set given marks of data in the frequency table and create a set of two rows and three columns in them with the marks listed in them.

Marks	Tally	Frequency
2           3           4           5           6           7           8           9          10

Step 2:

We can enter the highest mark of the students from the list in the table as follows.

Marks	Tally	Frequency
2            3            4            5            6            7            8            9           10	/              /              /

Step 3:

            We can find the student marks data of all information in the table. We can find the information of the frequency in the given marks of the students as shown.

Marks	Tally	Frequency
2              3              4              5              6              7              8              9             10	///             //             //             ////            //            ///            //           //            /	3             2             2             4             2             3              2              2               1

These are the brief on frequency table values that are on limits.

Percent Return Formula

In math, how much of parts done in every hundred is called as percents. The percents are represented by the symbol ‘%’. In other words, how much of value is noted out of hundred in experiments. The formula is returned with 100. Now we are going to see about percent return formula.

Explanations for Percents Return Formula in Math

Percents return formula:

                               The percents are represented as fraction with percentage symbol that is 32/100%. We can denote the percents in whole number also like 32%.T he formula for returns the percents are P = ( observed value / total value) x 100.   

How to return the percents using formula:

                             The formula for percents is divide the observed value and total value. Then multiply the 100 with that resultant value. Now, we can say this value is percents with symbol ‘%’. Sometimes, the formula returns the decimal value.   

                

How to returns the fraction into decimal value:

                               We can represent the percent value in fraction and if there is any possible, we can simplify the fraction. Then divide the numerator value with denominator value.

More about Percents Returns Formula

Example problems for percents return formula in math:

Problem 1: Return the percent value using formula for given expression.

The student got the marks 140 out of 200. What is the percent value of student?

Answer:

The percent return formula is P = (observed value / total value) x 100.

The observed value is 140.

Return the percent as (140/200) x 100 = 0.7 x 100 = 70%.

Therefore, the formula returns the percent value as 70%.

Problem 2: Return the percent value using formula for given expression.

The fruit seller has 1650 apples out of 300 fruits. What is the percent value of apple?

Answer:

The percent return formula is P = (observed value / total value) x 100.

The observed value is 165.

Return the percent as (165/300) x 100 = 0.55 x 100 = 55%.

Therefore, the formula returns the percent value as 55%.

Exercise problems for percents return formula:

1. Return the percent value using formula for 65/130.

Answer: The percent value is 50.

2. Return the percent value using formula for 87/150.

Answer: The percent value is 58.

Function Generator Application

Function is one of a concept in mathematics. Function is a relation such that every element of a known set is connected with an element of another set. A function generator is automatically generates the output value when we enter the input in a generator.  In online, few websites are providing function tutoring. In this article we shall discuss for used function generator.

Application of function generator:

Draw graph for the given functions

Evaluate the given functions

Find the output of given functions.                 

Sample Problem for Function Generator Application:

Function generator application problem 1:

Evaluate the given trigonometry functions f(x) = - sin x and make the graph for the given function.

Solution:

We are going to locate the points of the given equation and make the graph. In the first step we take f(x) as y, we get

       f(x) = -sin x

          y = -sin x

In the above equation we put x = -2

       y = -sin (-2)

       y = 0.909

In the above equation we put x = -1

       y = -sin (-1)

       y = 0.84

Like this we find out the plotting points. From the values we get the following values

X	-2	-1	0	1
y	0.909	0.84	0	-0.84

Graph:




Function Generator Application Problem 2:

Evaluate the given trigonometry functions f(x) = -cos x and make the graph for the given function.

Solution:
We are going to locate the points of the given equation and make the graph. In the first step we take f(x) as y, we get

       f(x) = -cos x

          y = -cos x

In the above equation we put x = -2

         y = -cos (-2)

         y = 0.41

In the above equation we put x = -1

         y = -cos (-1)

         y = -0.54

Like this we find out the plotting points. From the values we get the following values

X	-2	-1	0	1
y	0.41	-0.54	-1	-0.54

Graph:

Graph of Tanx

The tanx mean that the slope of function.that is tanx = sinx/cosx or rise/run.We know that tanx = sin x/cosx, tanx can be defined for all the values of x for which cosx ≠ 0.I.e. All the real numbers except odd integer multiples of π/2 (tanx is not obtained for cosx = 0 and hence not defined for x, an odd multiple of π/2).

More about Graph of Tanx:

Tangent Function Graph(y =tanx):

• In a graph of tanx 1 cycle takes place between the -90 degree to 90 degree.
• Each cycle contains are vertical asymptotes at the end.
• The period of tanx is `Pi`
• Tanx has no amplitude. That is graph that goes infinitely in vertical direction.
• The range(R) of the function tanx = ( - `oo` ,`oo`
• The domain of the function tanx is R - {(2k+1)`pi`
• The tanx is undefined at the value of x = 90° and x = 270°.
• The tanx graph crosses the x-axis three times in the interval
• 0° ≤ x ≤ 360°. At x = 0°, x = 180°, and x = 360°, tanx = 0.
• The maximum value of the function  tanx is `oo`
• The minimum value of the function  tanx is −`oo`

Consider the table for tanx between 0 to 180 degree

Graph:

The following graph shows the tanx function.

    


Problems Based on Graph of Tanx:

1.Sketch the graph for the function  y=10tant.

Graph:

Given y = 10 tant.

That is the radius of the circle is 10.The graph can be drawn by using the circle with radius 10.Let us see step by step process.





Problem 2:

Draw the graph for the function y = 25 tan t

Graph:



Practice Problem for Graph of Tanx:

1.Sketch the graph for the function y =17 tanx  by using the graphing calculator.

2.Sketch the graph for the function y =40 tanx  by using the graphing calculator.

3.Sketch the graph for the function y =33 tanx  by using the graphing calculator.

Derivatives Related Rates

Derivative related rates is the normal derivatives whereas the differentiation is carried out with respect to the time function t. The Differentiation of a given function with respect to time is called related rates derivatives. The related rates derivatives are also one of the parts of calculus which deals with calculating the rate of change of function with respect to time. The following are the example problems for related rates derivatives.

Related Rates Example Problems:

Example 1:

Find the related rate derivatives for the given function.

f(t) = t 4 – 18t + 16

Solution:

The given function is

f(t) = t 4 – 18t + 16

The first derivative f ' is given by

f '(t) = 4 t 3 – 18

Example 2:

Find the related rate derivatives for the given function.

f(t) = t 5 – 6 t 3  + 11

Solution:

The given function is

f(t) = t 5 – 6 t 3  + 10

The first derivative f ' is given by

f '(t) = 5t 4 – 6(3 t 2 )

f '(t) = 5t 4 – 18 t 2

Example 3:

Find the related rate derivatives for the given function.

f(t) = t2 – 4t + 8

Solution:

The given function is

f(t) = t 2 – 4t + 8

Differentiate the above equation with respect to t.

f '(t) = 2 t  – 4

Example 4:

Find the related rate derivatives for the given function.

f(t) = t 3 – 5 t 2  + 11t

Solution:

The given function is

f(t) = t 3 – 5 t 2  + 11t

The first derivative f ' is given by

f '(t) = 3t 2 – 5(2 t  ) + 11

f '(t) = 3t 2 – 10 t + 11

Example 5:

Find the related rate derivatives for the given function.

f(t) = t4 – 3t 3 – 4t 2  + t

Solution:

The given function is

f(t) = t4 – 3t 3 – 4 t 2  + t

The first derivative f ' is given by

f '(t) = 4 t 3 – 3(3t 2 ) – 4( 2 t  ) + 1

f '(t) = 4 t 3 – 9t 2  – 8 t  + 1

Related Rates Practice Problems:

1) Find the related rate derivatives for the given function.

f(t) = t 3 – 6 t 2  + 11t

Answer: f '(t) = 3t 2 – 12 t

2) Find the related rate derivatives for the given function.

f(t) = t 2 – 6 t   + 11

Answer: f '(t) = 2t – 6

Construct Probability Table

Construct Probability table is defined as an equation or table with its probability happenings. Usually, probability table is for constant occurrence for much number of data produces results. For construct probability table function we learn about discrete random variable, if a random variable obtains only a finite or a countable number of values, it is known as discrete random variable. To construct Probability table it contains probability mass function and moments.

Classification of Construct Probability Table:

Probability Mass Function:

The statistical description of discrete probability table function p (x)  functions and satisfies the following properties:

(1.) The probability which x can take a particular value x is p (x)

That is P(x = x) = p (x) = px.

(2). P(x) is a non – negative for every real x.

(3). The amount of p (x) over all likely values of x is one. Which is Σpi = 1 here j denotes that x and pi is the probability at x = xi

Moments:

A probable value of a function of a chance variable x is used for calculating the moments. Let us see the two types of moments.

(i) Moments about the origin.
(ii) Moments about the mean that are known as central moments.

Example Problem for Construct Probability Table:

A box has 4 green and 3 black pens. Construct a probability table distribution of number of black pens in 3 draws one by one from the box. (i) With replacement

Solution:

(i) With replacement

Let x be the approximate variable of drawing number of black pens in three draws.

X will takes the values 0,1,2,3.

P (Black pen) = `3 / 7 ` = P (B)

P (Not Black pen) = `4 / 7` = P (G)

Hence P(X = 0) = P (GGG) = `4/ 7` * `4 / 7` * `4 / 7` = `64 / 343`

P(X = 1) = P (BGG) + P (GBG) + P (GGB)

=  ( `3 / 7` * `3 / 7` * `4 / 7` )  + ( `4 / 7` * `3 / 7`   * `4 / 7` ) +  ( `4 / 7` * `3 / 7`   * `3 / 7` )

= 3 * (`48 / 343` )

= `144 / 343 `

P(X = 2) = P (BBG) + P (BGB) + P (GBB)

= (`3 / 7` * `3 / 7` * `4 / 7` ) + (`3 / 7` * `4 / 7` * `3 / 7` ) + (`4 / 7` * `3 / 7` * `3 / 7` )

= 3 * ( `3 / 7` ) * ( `3 / 7` ) * ( `4 / 7` )

= 3 * (`36 / 343` )

= `108 / 343`

P(X = 3) = P (BBB) = ( `3 / 7` ) * ( `3 / 7` ) * ( `3 / 7` )

= `27 / 343`

Therefore the wanted probability table distribution is

x	0	1	2	3
p (x =x)	`64 / 43`	`144 / 343`	`108 / 343`	`27 / 343`

Example 2:

Find out the probability mass function, and the collective distribution function for obtaining ‘3’s while two dice are thrown.

Solution:

2 dice are thrown. Let x be the approximate variable of obtaining number of ‘3’s. Hence x can take the values 0, 1, 2.

P (no ‘3’) = P (X = 0) =` 25 / 36`

P (one ‘3’) = P (X = 1) = `10 / 36`

P (two ‘3’s) = P (X = 2) = `1 / 36`

Obtained Probability mass function is

x	0	1	2
P (x = x)	`25 / 36`	`10 / 36`	`1 / 36`