Learning Natural Tangent Tables

The branch of math that deals with the relations of the sides and angles of triangles is known as trigonometry. Which the methods of reducing from a certain given parts to other required parts, and also in the general relations which exist between the trigonometrical functions of arcs or angles. The trigonometry ratio table is used to find the values of the trigonometry functions. Here we are going to see about learning of natural tangent tables:

The natural tangent of an angle is the ratio, the length of the opposite side to the length of the adjacent side. In our case

Tan A = `(opposite side) / (adjacent side)`

The natural tangent angle table will be very useful for finding the value of tangent angles. Learning of natural tangent tables will be very helpful when we are using the problems.


Learning Natural Tangent Tables:

Angle   Tangent(A)

Angle    Tangent (A)

0         0.000 1         0.017 2         0.035 3         0.052 4         0.070 5         0.087 6         0.105 7         0.123 8         0.141 9         0.158 10        0.176 11        0.194 12        0.213 13        0.231 14        0.249 15        0.268 16        0.287 17        0.306 18        0.325 19        0.344 20        0.364 21        0.384 22        0.404 23        0.424 24        0.445 25        0.466 26        0.488 27        0.510 28        0.532 29        0.554 30        0.577 31        0.601 32        0.625 33        0.649 34        0.675 35        0.700 36        0.727 37        0.754 38        0.781 39        0.810 40        0.839 41        0.869 42        0.900 43        0.933 44        0.966 45        1.000

45        1.000 46        1.036 47        1.072 48        1.111 49        1.150 50        1.192 51        1.235 52        1.280 53        1.327 54        1.376 55        1.428 56        1.483 57        1.540 58        1.600 59        1.664 60        1.732 61        1.804 62        1.881 63        1.963 64        2.050 65        2.145 66        2.246 67        2.356 68        2.475 69        2.605 70        2.747 71        2.904 72        3.078 73        3.271 74        3.487 75        3.732 76        4.011 77        4.331 78        4.705 79        5.145 80        5.671 81        6.314 82        7.115 83        8.144 84        9.514 85        11.430 86        14.301 87        19.081 88        28.636 89

Examples for Learning Natural Tangent Tables:

Find the values of the given natural tangent angles using natural tangent table.

1) tangent 1   =      0.017

2) tangent 48   =   1.1114

3) tangent 9   =     0.158

4) tangent 56  =     1.483

5) tangent 16   =     0.287

6) tangent 64   =     2.050

7) tangent 25    =    0.466

8) tangent 78    =    4.705

9) tangent 44   =    0.966

10) tangent 88   =    28.636

Multiplication tables

In this article we are going to see about multiplication tables.  Multiplication facts table lets you learn and practice the basic time tables using the numbers zero through twelve. Children are no longer memorizing their multiplication facts table. There are many tricks to teach children multiplication facts table in mathematics.

Simplest Method for Multiplication Facts Table

Remember, work on the 2's, 5's and 10's  first; then the doubles (6x6, 7 x7, 8x8); then move to each of the fact families: 3's, 4,s, 6's, 7's, 8's and 9's, 11's and 12's. Don’t move to a different fact family without mastering the previous one.

Simplest method for multiplication facts table:

x	0	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10	11	12
2	0	2	4	6	8	10	12	14	16	18	20	22	24
3	0	3	6	9	12	15	18	21	24	27	30	33	36
4	0	4	8	12	16	20	24	28	32	40	40	44	48
5	0	5	10	15	20	25	30	35	40	45	50	55	60
6	0	6	12	18	24	30	36	42	48	54	60	66	72
7	0	7	14	21	28	35	42	49	56	63	70	77	84
9	0	8	16	24	32	40	48	56	64	72	80	88	96
9	0	9	18	27	36	45	54	63	72	81	90	99	108
10	0	10	20	30	40	50	60	70	80	90	100	110	120
11	0	11	22	33	44	55	66	77	88	99	110	121	132
12	0	12	24	36	48	60	72	84	96	108	120	132	144

Your life will be lot easier when you can simply remember the multiplication facts table. Please train your memory.  First, you can use the multiplication facts table above to start putting the answers into your memory. Use it a few times per day for about 10 minutes each, and you will learn your multiplication tables.

Graph of Y Square Root of X

Generally a function is described in the form y = f(x), y being the value the function at any general point x in the domain of the function. The letter y is chosen because the values of the function is represented by the y-axis in a graph.

The graph of the function gives a visual presentation of the function which helps us to study the function at a glance.

The values of square root of positive real numbers is expressed as function f(x) = square root of x or simply y = square root of x. The graph of y = square root of x is an interesting study and let us take a closer look.


Concept of a Square Root Function:

Consider a simple square function y = x2. It is a special  form of a quadratic function. We know that the graph of a quadratic function is a parabola. In this case it is a vertical, concave up parabola with origin as vertex. The graph is shown below.



Let see what is the inverse of the function y = x2

y = x2, or, √y = x and by interchanging the variable we get the inverse function as,

y = √x

Thus the square root function is nothing but the inverse of the square function.

We had seen that the graph of y = = x2 is symmetrical over y –axis. Therefore, the graph of y = square root of x has to be symmetrical over x-axis. However, since negative values of root of x have no meaning, the graph exists only in the first quadrant of the Cartesian system. It is shown below.




Example Problems on Graph of Y Square Root of X

From the graph of y square root of x, find the value of square root 3 to the nearest tenths. (Assume the sale to be 1unit on both the axes)



Since the scale of the graph is 1 unit each small division represents 0.2 unit. It is seen from the graph, when x = 5, the value of y is 1.7. Hence, square root of 3 is 1.7, nearest to tenths.

Use the graph of y square root of x, to evaluate 2( square root 6) to the nearest tenths. (Assume the sale to be 1unit on both the axes)



Since the scale of the graph is 1 unit each small division represents 0.2 unit. It is seen from the graph, when x = 6, the value of y is 2.5. Hence, 2(square root of 6) = 2(2.5) = 5

Solve Cumulative Probability Distribution Function

Computation of probability of events is called as probability distribution.  It can give all possible outputs in a single process.  Cumulative probability distribution function of a random variable X is defined as F(x) = P(X <= x) = sum_(x)I <= x)p(x_i): (-oo < x oo).  Let us discuss about the topic of “solve cumulative probability distribution function” in the following below with some related problems.

Example Problems – Solve Cumulative Probability Distribution Function

Example problem 1

A random variable Y has the following probability mass function.  Find the value of b and evaluate `P(Y lt= 2)` .

Y	0	1	2	3	4
P(Y = y)	10b	20b	30b	40b	50b

Solution:

P(Y = y) is the probability mass function `sum_(y = 0)^4 (Y = y) = 1`

P(Y=0) + P(Y = 1) + P(Y = 2) + P(Y = 3) + P(Y = 4) = 1

10b + 20b + 30b + 40b + 50b = 1

150b = 1

b = `1/150`

`P(Y lt= 2) = P(Y = 0) + P(Y = 1)`

                = `10/150 + 20/150 + 30/150`

                = `(10 + 20 + 30)/150`

                = `60/150`

                = `2/5`

Answer:

The value of b is `1/150`

`P(Y lt= 2) = 2/5`

Example problem 2

Find out probability mass function and cumulative distribution function for getting ‘4’s when two dice are thrown.

Solution:

Let us consider Y be the random variable of getting number of ‘4’s.

First, we are going to find the probability mass function

Therefore, Y can take the values 0, 1, 2.

P(no ‘4’) = P(Y = 0) = `25/36`

P(one ‘4’) = P(Y = 1) = `10/36`

P(two ‘4’) = P(Y = 2) = `1/36`

Probability mass function

y	0	1	2
P(Y = y)	25/36	10/36	1/36

Now, we are going to find the cumulative probability distribution function

Formula:  `F(x) = sum_(x_i = -oo)^x P(X = x_i)`

Y = 0, F(0) = P(Y = 0) = 25/36

Y = 1, `F(1) = sum_(i = -oo)^1 P(Y = y_i)`

                = P(Y = 0) + P(Y = 1)

                = `25/36 + 10/36`

                = `35/36`

Y = 2, `F(2) = sum_(i = -oo)^2 P(Y = y_i)`

                = P(Y = 0) + P(Y = 1) + P(Y = 2)

                = `25/36 + 10/36 + 1/36`

                = `36/36`

                = 1

Cumulative distribution function

y	0	1	2
F(Y)	25/36	35/36	1

Answer:

Probability mass function

y	0	1	2
P(Y = y)	25/36	10/36	1/36

Cumulative distribution function

y	0	1	2
F(Y)	25/36	35/36	1

Practicing Problems – Solve Cumulative Probability Distribution Function

Practicing problem 1

Find the cumulative distribution function for the given probability mass function

y	0	1	2	3
P(Y = y)	2a	4a	6a	8a

Find the value of a and `P(Y lt= 3)`

Answer:

The value of a is `1/20`

`P(Y lt= 3) = 1`

Practicing problem 2

Find the cumulative distribution function for getting number of heads when three coins are tossed once.

Answer:

Cumulative distribution function

y	0	1	2	3
F(y)	1/8	1/2	7/8	1

Triangle Geometry Cosine

Look at this geometry figure:



In all these right angled triangles of different sizes, <A has the same measure. Since <B = 90o, we have <C = 90o - <A, in each of these triangles. So, all these triangles have equal angles and hence are similar. Thus even though the lengths of the sides BC, CA, AB are different in these, the numbers `(BC)/(AC)` and `(AB)/(AC)` are the same for all these triangles.

In all these triangles, AC is the hypotenuse. The side BC is the opposite side of <A. The third side AB is called the adjacent side of<A.    In a right angled triangle with <A as one of its angles, the number obtained by dividing the adjacent side of  <A by the hypotenuse is called the cosine of <A and its written cos A. Thus

cos A = `(adjacent- side- of- angleA) / ( hypotenuse )`

Note that cos A is a number indicating the size of <A alone and not depend on the size of the right angled triangle containing <A.

Examples for Triangle Geometry Cosine:

Example 1:

Determine the cosine of angle A or cos A; If <A =45o in the right angled triangle ABC, as in the figure below



then, as seen earlier, we have

`(BC)/(AC)` = `(AB )/(AC)` =1/`sqrt(2)`

which in our new terminology can be written

cos 45o = 1/`sqrt(2)`

Example 2:

Determine the cos30o, if <A =30o in the right angled triangle ABC, as in the figure below



Solution:

By definition, we know that   cos A = `(AB)/(AC)`

Therefore,   cos 30 = `sqrt(3)` / 2


Example 3:

Measure the cos C, if <C =60o in the right angled triangle ABC, as in the figure below





Solution:

Given:

`Delta` ABC , AB = `sqrt(3)` , BC = 1 and AC = 2.

To find the triangle geometry cosine C:

By definition, we know that,  cos C = `(BC)/(AC)`

That is                                     cos 60o = `1/2`

This is the required triangle geometry cosine values.

Example 4:

Measure the cosine of angle A or cos A; If <A =20o, AB = 4.7 and AC =5 in the right angled triangle ABC, as in the figure below



Solution:

Given:

In `Delta` ABC, <A = 20o , AB = 4.7 and AC =5.

To find the triangle geometry cosine of angle A:

We have found by actual measurement that in such a triangle as `Delta` ABC

`(AB)/(AC)` = `(4.7) /(5)`  = 0.94

This means in other words

cos 20o = 0.94