Isosceles Triangle

Isosceles triangle is a three sided closed polygon with two equal sides and two equal angles.

Propeties of an isosceles triangle:

The legs are equal in the length and the third side is the base of the triangle.
The angles formed in the base side are equal in measures.i.e. they they are congruent to each other.
The angle formed by two legs of the triangle is vertex angle.

geometry isoscles triangle

Formula related related to isosceles triangle:

1. Formula to find the vertex angle of the isosceles triangle is

Sum of the angles in triangle is 180 degrees.

Here the base angle is equal.

a+a+ vertex angle= 180

Vertex angle = 180-2a.

2.Formula for the area of the isosceles triangle is `1/2 (BH)` square.units.

3.Formula for the perimeter of the isosceles triangle is sum of three sides.

Model problems to isosceles triangle

Pro 1: Find the vertex angle of the triangle whose base angle is 70 degrees?

Sol: Step 1:   Let the base angle be 70 degrees

We know that the base angles in the isosceles traingle is equal

Step 2:  Sum of the angles in triangle is 180 degrees.

Angle 1+ Angle 2+ Angle 3 = 180 degrees

x + x + vertex angle = 180

Step 3:   Vertex angle= 180- (140)

Vertex angle is 40 degrees

Vertex angle of the geometry isosceles triangle is 40 degrees.

Pro 2. Find the area of the isosceles triangle whose base side is 10cm and the height of the triangle is 3 cm.

Sol:  Step 1:   Base side, B= 10cm

Height of the triangle, H= 3cm

Step 2:  Formula for the area of the isosceles triangle is 1/2 (BH) square.units.

= `(1/2) (10 xx3)`

= `(30/2)`

= 15cm2

Area of the isosceles triangle = 15cm2

Pro 3: Find the perimeter of the isosceles triangle whose base side is 10cm and leg side is 5cm?

Sol: Step 1: We know that leg sides are equal in length.

Perimeter of the isosceles triangle  = (10+5+5)

= 20 cm

Perimeter of the isosceles triangle is 20cm

Geometry Reflections

In geometric, the determination of reflection of a point in a line translation is performed only through a given point or line. In reflection of a point in a line, the basic ideas of reflection are the transformation of a point to a reflected point that is the equal length of the opposite side of a  line. The transformation of  reflections in two right angular axes produce a rotation of straight angle (180°), that is a half turn.

Reflection of a point:

Reflection:

The reflection is the "flip or mirror image" of it over a line.  So, the actual and mirror images are identical to each other.

The two very common reflections of polygons are given by

Horizontal reflections.

Vertical reflections.

The colored vertices used for each of the triangle.  The line of reflection is halfway from both red points, blue points, and green points. The line of reflection is directly in the center of both points.

Reflection of  a point in a line to describe the X-axis is consider the mirror line or axis of reflection. Therefore, we change the given geometric object point  into x = x  and y = -y.

geometry reflections

Reflection of  a point in a line to describe the Y-axis is consider the mirror line or axis of reflection.. Therefore, we change the given geometric object point  into x = -x  and y = y.

geometry reflections

Reflection Example:

We take a shape of triangle, Learn horizontal reflection assists, the transformation of the geometric triangle of co-ordinates are P, Q, and R. It is transformed by using the horizontal reflection, that is the horizontal line y = x.

Solution:

Step 1: To draw the triangle object from the given co-ordinates are P, Q, and R in the X-Y plane.

geometry reflections

Step 2:  To draw the horizontal line or mirror line of that triangle PQR, and draw the perpendicular line to the horizontal line or mirror line from each co-ordinates of  given triangular object. And then measure the distance of each perpendicular lines are y1, y2, y3.

geometry reflections

Step 3:  To plot the mirrored points are P', Q', R', the same vertical distance from the horizontal line or mirror line.

geometry reflections

Step 4:  To join the plotted points P', Q', R'. We get the horizontally reflected triangular object.

geometry reflections

Step 5:   This is required horizontal reflections of triangular object.

Horizontal reflection of X- axis:

Horizontal reflection assist that concepts to describe the x-axis are considering the mirror line. Then change all the coordinates of the given geometric object into the x = x and y = -y. For example,

geometry reflections

Common Logarithm Table

A numerical expressipon may involve multiplication, division, or exponents of large numbers. In all such calculations, the log tables prove to be an effective tool in simplyfying the complicated computations. Before seeing the common logarithm table, let us learn what characteristic and mantissa is.

In general, the logarithm of any number "n" can be expresed as follows:
log n = p + log m, where p is an integer and 1 ≤ m < 10.
"p" is the integral of log n, is defined as the characteristic and th elog m is defined as mantissa.
Since log 1 = 0 and log 10 = 1 and 1 ≤ m < 10, we have log m as a decimal.
Note: The charcteristic of a number greater than 1 is one less than the number of digits in its integral part.

Ex: Charcteristic of 36177 is 4
Total number of digits = 5, one less than 5 is 4.
Consider log 36.122
We have only two digits in the integral part. So, characteristic of 36.122 is 1.

Common logarithm table from 1 to 4.99

1	0	1.1	0.0413927	2.1	0.3222193	3.1	0.4913617	4.1	0.6127839
1.001	0.0004341	1.11	0.045323	2.11	0.3242825	3.11	0.4927604	4.11	0.6138418
1.002	0.0008677	1.12	0.049218	2.12	0.3263359	3.12	0.4941546	4.12	0.6148972
1.003	0.0013009	1.13	0.0530784	2.13	0.3283796	3.13	0.4955443	4.13	0.6159501
1.004	0.0017337	1.14	0.0569049	2.14	0.3304138	3.14	0.4969296	4.14	0.6170003
1.005	0.0021661	1.15	0.0606978	2.15	0.3324385	3.15	0.4983106	4.15	0.6180481
1.006	0.002598	1.16	0.064458	2.16	0.3344538	3.16	0.4996871	4.16	0.6190933
1.007	0.0030295	1.17	0.0681859	2.17	0.3364597	3.17	0.5010593	4.17	0.6201361
1.008	0.0034605	1.18	0.071882	2.18	0.3384565	3.18	0.5024271	4.18	0.6211763
1.009	0.0038912	1.19	0.075547	2.19	0.3404441	3.19	0.5037907	4.19	0.622214
1.01	0.0043214	1.2	0.0791812	2.2	0.3424227	3.2	0.50515	4.2	0.6232493
1.011	0.0047512	1.21	0.0827854	2.21	0.3443923	3.21	0.506505	4.21	0.6242821
1.012	0.0051805	1.22	0.0863598	2.22	0.346353	3.22	0.5078559	4.22	0.6253125
1.013	0.0056095	1.23	0.0899051	2.23	0.3483049	3.23	0.5092025	4.23	0.6263404
1.014	0.006038	1.24	0.0934217	2.24	0.350248	3.24	0.510545	4.24	0.6273659
1.015	0.006466	1.25	0.09691	2.25	0.3521825	3.25	0.5118834	4.25	0.6283889
1.016	0.0068937	1.26	0.1003705	2.26	0.3541084	3.26	0.5132176	4.26	0.6294096
1.017	0.007321	1.27	0.1038037	2.27	0.3560259	3.27	0.5145478	4.27	0.6304279
1.018	0.0077478	1.28	0.10721	2.28	0.3579348	3.28	0.5158738	4.28	0.6314438
1.019	0.0081742	1.29	0.1105897	2.29	0.3598355	3.29	0.5171959	4.29	0.6324573
1.02	0.0086002	1.3	0.1139434	2.3	0.3617278	3.3	0.5185139	4.3	0.6334685
1.021	0.0090257	1.31	0.1172713	2.31	0.363612	3.31	0.519828	4.31	0.6344773
1.022	0.0094509	1.32	0.1205739	2.32	0.365488	3.32	0.5211381	4.32	0.6354837
1.023	0.0098756	1.33	0.1238516	2.33	0.3673559	3.33	0.5224442	4.33	0.6364879
1.024	0.0103	1.34	0.1271048	2.34	0.3692159	3.34	0.5237465	4.34	0.6374897
1.025	0.0107239	1.35	0.1303338	2.35	0.3710679	3.35	0.5250448	4.35	0.6384893
1.026	0.0111474	1.36	0.1335389	2.36	0.372912	3.36	0.5263393	4.36	0.6394865
1.027	0.0115704	1.37	0.1367206	2.37	0.3747483	3.37	0.5276299	4.37	0.6404814
1.028	0.0119931	1.38	0.1398791	2.38	0.376577	3.38	0.5289167	4.38	0.6414741
1.029	0.0124154	1.39	0.1430148	2.39	0.3783979	3.39	0.5301997	4.39	0.6424645
1.03	0.0128372	1.4	0.146128	2.4	0.3802112	3.4	0.5314789	4.4	0.6434527
1.031	0.0132587	1.41	0.1492191	2.41	0.382017	3.41	0.5327544	4.41	0.6444386
1.032	0.0136797	1.42	0.1522883	2.42	0.3838154	3.42	0.5340261	4.42	0.6454223
1.033	0.0141003	1.43	0.155336	2.43	0.3856063	3.43	0.5352941	4.43	0.6464037
1.034	0.0145205	1.44	0.1583625	2.44	0.3873898	3.44	0.5365584	4.44	0.647383
1.035	0.0149404	1.45	0.161368	2.45	0.3891661	3.45	0.5378191	4.45	0.64836
1.036	0.0153598	1.46	0.1643529	2.46	0.3909351	3.46	0.5390761	4.46	0.6493349
1.037	0.0157788	1.47	0.1673173	2.47	0.392697	3.47	0.5403295	4.47	0.6503075
1.038	0.0161974	1.48	0.1702617	2.48	0.3944517	3.48	0.5415792	4.48	0.651278
1.039	0.0166156	1.49	0.1731863	2.49	0.3961993	3.49	0.5428254	4.49	0.6522463
1.04	0.0170333	1.5	0.1760913	2.5	0.39794	3.5	0.544068	4.5	0.6532125
1.041	0.0174507	1.51	0.1789769	2.51	0.3996737	3.51	0.5453071	4.51	0.6541765
1.042	0.0178677	1.52	0.1818436	2.52	0.4014005	3.52	0.5465427	4.52	0.6551384
1.043	0.0182843	1.53	0.1846914	2.53	0.4031205	3.53	0.5477747	4.53	0.6560982
1.044	0.0187005	1.54	0.1875207	2.54	0.4048337	3.54	0.5490033	4.54	0.6570559
1.045	0.0191163	1.55	0.1903317	2.55	0.4065402	3.55	0.5502284	4.55	0.6580114
1.046	0.0195317	1.56	0.1931246	2.56	0.40824	3.56	0.55145	4.56	0.6589648
1.047	0.0199467	1.57	0.1958997	2.57	0.4099331	3.57	0.5526682	4.57	0.6599162
1.048	0.0203613	1.58	0.1986571	2.58	0.4116197	3.58	0.553883	4.58	0.6608655
1.049	0.0207755	1.59	0.2013971	2.59	0.4132998	3.59	0.5550944	4.59	0.6618127
1.05	0.0211893	1.6	0.20412	2.6	0.4149733	3.6	0.5563025	4.6	0.6627578
1.051	0.0216027	1.61	0.2068259	2.61	0.4166405	3.61	0.5575072	4.61	0.6637009
1.052	0.0220157	1.62	0.209515	2.62	0.4183013	3.62	0.5587086	4.62	0.664642
1.053	0.0224284	1.63	0.2121876	2.63	0.4199557	3.63	0.5599066	4.63	0.665581
1.054	0.0228406	1.64	0.2148438	2.64	0.4216039	3.64	0.5611014	4.64	0.666518
1.055	0.0232525	1.65	0.2174839	2.65	0.4232459	3.65	0.5622929	4.65	0.667453
1.056	0.0236639	1.66	0.2201081	2.66	0.4248816	3.66	0.5634811	4.66	0.6683859
1.057	0.024075	1.67	0.2227165	2.67	0.4265113	3.67	0.5646661	4.67	0.6693169
1.058	0.0244857	1.68	0.2253093	2.68	0.4281348	3.68	0.5658478	4.68	0.6702459
1.059	0.024896	1.69	0.2278867	2.69	0.4297523	3.69	0.5670264	4.69	0.6711728
1.06	0.0253059	1.7	0.2304489	2.7	0.4313638	3.7	0.5682017	4.7	0.6720979
1.061	0.0257154	1.71	0.2329961	2.71	0.4329693	3.71	0.5693739	4.71	0.6730209
1.062	0.0261245	1.72	0.2355284	2.72	0.4345689	3.72	0.5705429	4.72	0.673942
1.063	0.0265333	1.73	0.2380461	2.73	0.4361626	3.73	0.5717088	4.73	0.6748611
1.064	0.0269416	1.74	0.2405492	2.74	0.4377506	3.74	0.5728716	4.74	0.6757783
1.065	0.0273496	1.75	0.243038	2.75	0.4393327	3.75	0.5740313	4.75	0.6766936
1.066	0.0277572	1.76	0.2455127	2.76	0.4409091	3.76	0.5751878	4.76	0.677607
1.067	0.0281644	1.77	0.2479733	2.77	0.4424798	3.77	0.5763414	4.77	0.6785184
1.068	0.0285713	1.78	0.25042	2.78	0.4440448	3.78	0.5774918	4.78	0.6794279
1.069	0.0289777	1.79	0.252853	2.79	0.4456042	3.79	0.5786392	4.79	0.6803355
1.07	0.0293838	1.8	0.2552725	2.8	0.447158	3.8	0.5797836	4.8	0.6812412
1.071	0.0297895	1.81	0.2576786	2.81	0.4487063	3.81	0.580925	4.81	0.6821451
1.072	0.0301948	1.82	0.2600714	2.82	0.4502491	3.82	0.5820634	4.82	0.683047
1.073	0.0305997	1.83	0.2624511	2.83	0.4517864	3.83	0.5831988	4.83	0.6839471
1.074	0.0310043	1.84	0.2648178	2.84	0.4533183	3.84	0.5843312	4.84	0.6848454
1.075	0.0314085	1.85	0.2671717	2.85	0.4548449	3.85	0.5854607	4.85	0.6857417
1.076	0.0318123	1.86	0.2695129	2.86	0.456366	3.86	0.5865873	4.86	0.6866363
1.077	0.0322157	1.87	0.2718416	2.87	0.4578819	3.87	0.587711	4.87	0.687529
1.078	0.0326188	1.88	0.2741578	2.88	0.4593925	3.88	0.5888317	4.88	0.6884198
1.079	0.0330214	1.89	0.2764618	2.89	0.4608978	3.89	0.5899496	4.89	0.6893089
1.08	0.0334238	1.9	0.2787536	2.9	0.462398	3.9	0.5910646	4.9	0.6901961
1.081	0.0338257	1.91	0.2810334	2.91	0.463893	3.91	0.5921768	4.91	0.6910815
1.082	0.0342273	1.92	0.2833012	2.92	0.4653829	3.92	0.5932861	4.92	0.6919651
1.083	0.0346285	1.93	0.2855573	2.93	0.4668676	3.93	0.5943926	4.93	0.6928469
1.084	0.0350293	1.94	0.2878017	2.94	0.4683473	3.94	0.5954962	4.94	0.6937269
1.085	0.0354297	1.95	0.2900346	2.95	0.469822	3.95	0.5965971	4.95	0.6946052
1.086	0.0358298	1.96	0.2922561	2.96	0.4712917	3.96	0.5976952	4.96	0.6954817
1.087	0.0362295	1.97	0.2944662	2.97	0.4727564	3.97	0.5987905	4.97	0.6963564
1.088	0.0366289	1.98	0.2966652	2.98	0.4742163	3.98	0.5998831	4.98	0.6972293
1.089	0.0370279	1.99	0.2988531	2.99	0.4756712	3.99	0.6009729	4.99	0.6981005
1.09	0.0374265	2	0.30103	3	0.4771213	4	0.60206
1.091	0.0378248	2.01	0.3031961	3.01	0.4785665	4.01	0.6031444
1.092	0.0382226	2.02	0.3053514	3.02	0.4800069	4.02	0.6042261
1.093	0.0386202	2.03	0.307496	3.03	0.4814426	4.03	0.605305
1.094	0.0390173	2.04	0.3096302	3.04	0.4828736	4.04	0.6063814
1.095	0.0394141	2.05	0.3117539	3.05	0.4842998	4.05	0.607455
1.096	0.0398106	2.06	0.3138672	3.06	0.4857214	4.06	0.608526
1.097	0.0402066	2.07	0.3159703	3.07	0.4871384	4.07	0.6095944
1.098	0.0406023	2.08	0.3180633	3.08	0.4885507	4.08	0.6106602
1.099	0.0409977	2.09	0.3201463	3.09	0.4899585	4.09	0.6117233

Common logarithm table from 5 to 9.99

5	0.69897	6	0.7781513	7	0.845098	8	0.90309	9	0.9542425
5.01	0.6998377	6.01	0.7788745	7.01	0.845718	8.01	0.9036325	9.01	0.9547248
5.02	0.7007037	6.02	0.7795965	7.02	0.8463371	8.02	0.9041744	9.02	0.9552065
5.03	0.701568	6.03	0.7803173	7.03	0.8469553	8.03	0.9047155	9.03	0.9556878
5.04	0.7024305	6.04	0.7810369	7.04	0.8475727	8.04	0.905256	9.04	0.9561684
5.05	0.7032914	6.05	0.7817554	7.05	0.8481891	8.05	0.9057959	9.05	0.9566486
5.06	0.7041505	6.06	0.7824726	7.06	0.8488047	8.06	0.906335	9.06	0.9571282
5.07	0.705008	6.07	0.7831887	7.07	0.8494194	8.07	0.9068735	9.07	0.9576073
5.08	0.7058637	6.08	0.7839036	7.08	0.8500333	8.08	0.9074114	9.08	0.9580858
5.09	0.7067178	6.09	0.7846173	7.09	0.8506462	8.09	0.9079485	9.09	0.9585639
5.1	0.7075702	6.1	0.7853298	7.1	0.8512583	8.1	0.908485	9.1	0.9590414
5.11	0.7084209	6.11	0.7860412	7.11	0.8518696	8.11	0.9090209	9.11	0.9595184
5.12	0.70927	6.12	0.7867514	7.12	0.85248	8.12	0.909556	9.12	0.9599948
5.13	0.7101174	6.13	0.7874605	7.13	0.8530895	8.13	0.9100905	9.13	0.9604708
5.14	0.7109631	6.14	0.7881684	7.14	0.8536982	8.14	0.9106244	9.14	0.9609462
5.15	0.7118072	6.15	0.7888751	7.15	0.854306	8.15	0.9111576	9.15	0.9614211
5.16	0.7126497	6.16	0.7895807	7.16	0.854913	8.16	0.9116902	9.16	0.9618955
5.17	0.7134905	6.17	0.7902852	7.17	0.8555192	8.17	0.9122221	9.17	0.9623693
5.18	0.7143298	6.18	0.7909885	7.18	0.8561244	8.18	0.9127533	9.18	0.9628427
5.19	0.7151674	6.19	0.7916906	7.19	0.8567289	8.19	0.9132839	9.19	0.9633155
5.2	0.7160033	6.2	0.7923917	7.2	0.8573325	8.2	0.9138139	9.2	0.9637878
5.21	0.7168377	6.21	0.7930916	7.21	0.8579353	8.21	0.9143432	9.21	0.9642596
5.22	0.7176705	6.22	0.7937904	7.22	0.8585372	8.22	0.9148718	9.22	0.9647309
5.23	0.7185017	6.23	0.794488	7.23	0.8591383	8.23	0.9153998	9.23	0.9652017
5.24	0.7193313	6.24	0.7951846	7.24	0.8597386	8.24	0.9159272	9.24	0.965672
5.25	0.7201593	6.25	0.79588	7.25	0.860338	8.25	0.9164539	9.25	0.9661417
5.26	0.7209857	6.26	0.7965743	7.26	0.8609366	8.26	0.91698	9.26	0.966611
5.27	0.7218106	6.27	0.7972675	7.27	0.8615344	8.27	0.9175055	9.27	0.9670797
5.28	0.7226339	6.28	0.7979596	7.28	0.8621314	8.28	0.9180303	9.28	0.967548
5.29	0.7234557	6.29	0.7986506	7.29	0.8627275	8.29	0.9185545	9.29	0.9680157
5.3	0.7242759	6.3	0.7993405	7.3	0.8633229	8.3	0.9190781	9.3	0.9684829
5.31	0.7250945	6.31	0.8000294	7.31	0.8639174	8.31	0.919601	9.31	0.9689497
5.32	0.7259116	6.32	0.8007171	7.32	0.8645111	8.32	0.9201233	9.32	0.9694159
5.33	0.7267272	6.33	0.8014037	7.33	0.865104	8.33	0.920645	9.33	0.9698816
5.34	0.7275413	6.34	0.8020893	7.34	0.8656961	8.34	0.9211661	9.34	0.9703469
5.35	0.7283538	6.35	0.8027737	7.35	0.8662873	8.35	0.9216865	9.35	0.9708116
5.36	0.7291648	6.36	0.8034571	7.36	0.8668778	8.36	0.9222063	9.36	0.9712758
5.37	0.7299743	6.37	0.8041394	7.37	0.8674675	8.37	0.9227255	9.37	0.9717396
5.38	0.7307823	6.38	0.8048207	7.38	0.8680564	8.38	0.923244	9.38	0.9722028
5.39	0.7315888	6.39	0.8055009	7.39	0.8686444	8.39	0.923762	9.39	0.9726656
5.4	0.7323938	6.4	0.80618	7.4	0.8692317	8.4	0.9242793	9.4	0.9731279
5.41	0.7331973	6.41	0.806858	7.41	0.8698182	8.41	0.924796	9.41	0.9735896
5.42	0.7339993	6.42	0.807535	7.42	0.8704039	8.42	0.9253121	9.42	0.9740509
5.43	0.7347998	6.43	0.808211	7.43	0.8709888	8.43	0.9258276	9.43	0.9745117
5.44	0.7355989	6.44	0.8088859	7.44	0.8715729	8.44	0.9263424	9.44	0.974972
5.45	0.7363965	6.45	0.8095597	7.45	0.8721563	8.45	0.9268567	9.45	0.9754318
5.46	0.7371926	6.46	0.8102325	7.46	0.8727388	8.46	0.9273704	9.46	0.9758911
5.47	0.7379873	6.47	0.8109043	7.47	0.8733206	8.47	0.9278834	9.47	0.97635
5.48	0.7387806	6.48	0.811575	7.48	0.8739016	8.48	0.9283959	9.48	0.9768083
5.49	0.7395723	6.49	0.8122447	7.49	0.8744818	8.49	0.9289077	9.49	0.9772662
5.5	0.7403627	6.5	0.8129134	7.5	0.8750613	8.5	0.9294189	9.5	0.9777236
5.51	0.7411516	6.51	0.813581	7.51	0.8756399	8.51	0.9299296	9.51	0.9781805
5.52	0.7419391	6.52	0.8142476	7.52	0.8762178	8.52	0.9304396	9.52	0.9786369
5.53	0.7427251	6.53	0.8149132	7.53	0.876795	8.53	0.930949	9.53	0.9790929
5.54	0.7435098	6.54	0.8155777	7.54	0.8773713	8.54	0.9314579	9.54	0.9795484
5.55	0.744293	6.55	0.8162413	7.55	0.877947	8.55	0.9319661	9.55	0.9800034
5.56	0.7450748	6.56	0.8169038	7.56	0.8785218	8.56	0.9324738	9.56	0.9804579
5.57	0.7458552	6.57	0.8175654	7.57	0.8790959	8.57	0.9329808	9.57	0.9809119
5.58	0.7466342	6.58	0.8182259	7.58	0.8796692	8.58	0.9334873	9.58	0.9813655
5.59	0.7474118	6.59	0.8188854	7.59	0.8802418	8.59	0.9339932	9.59	0.9818186
5.6	0.748188	6.6	0.8195439	7.6	0.8808136	8.6	0.9344985	9.6	0.9822712
5.61	0.7489629	6.61	0.8202015	7.61	0.8813847	8.61	0.9350032	9.61	0.9827234
5.62	0.7497363	6.62	0.820858	7.62	0.881955	8.62	0.9355073	9.62	0.9831751
5.63	0.7505084	6.63	0.8215135	7.63	0.8825245	8.63	0.9360108	9.63	0.9836263
5.64	0.7512791	6.64	0.8221681	7.64	0.8830934	8.64	0.9365137	9.64	0.984077
5.65	0.7520484	6.65	0.8228216	7.65	0.8836614	8.65	0.9370161	9.65	0.9845273
5.66	0.7528164	6.66	0.8234742	7.66	0.8842288	8.66	0.9375179	9.66	0.9849771
5.67	0.7535831	6.67	0.8241258	7.67	0.8847954	8.67	0.9380191	9.67	0.9854265
5.68	0.7543483	6.68	0.8247765	7.68	0.8853612	8.68	0.9385197	9.68	0.9858754
5.69	0.7551123	6.69	0.8254261	7.69	0.8859263	8.69	0.9390198	9.69	0.9863238
5.7	0.7558749	6.7	0.8260748	7.7	0.8864907	8.7	0.9395193	9.7	0.9867717
5.71	0.7566361	6.71	0.8267225	7.71	0.8870544	8.71	0.9400182	9.71	0.9872192
5.72	0.757396	6.72	0.8273693	7.72	0.8876173	8.72	0.9405165	9.72	0.9876663
5.73	0.7581546	6.73	0.8280151	7.73	0.8881795	8.73	0.9410142	9.73	0.9881128
5.74	0.7589119	6.74	0.8286599	7.74	0.888741	8.74	0.9415114	9.74	0.988559
5.75	0.7596678	6.75	0.8293038	7.75	0.8893017	8.75	0.9420081	9.75	0.9890046
5.76	0.7604225	6.76	0.8299467	7.76	0.8898617	8.76	0.9425041	9.76	0.9894498
5.77	0.7611758	6.77	0.8305887	7.77	0.890421	8.77	0.9429996	9.77	0.9898946
5.78	0.7619278	6.78	0.8312297	7.78	0.8909796	8.78	0.9434945	9.78	0.9903389
5.79	0.7626786	6.79	0.8318698	7.79	0.8915375	8.79	0.9439889	9.79	0.9907827
5.8	0.763428	6.8	0.8325089	7.8	0.8920946	8.8	0.9444827	9.8	0.9912261
5.81	0.7641761	6.81	0.8331471	7.81	0.892651	8.81	0.9449759	9.81	0.991669
5.82	0.764923	6.82	0.8337844	7.82	0.8932068	8.82	0.9454686	9.82	0.9921115
5.83	0.7656686	6.83	0.8344207	7.83	0.8937618	8.83	0.9459607	9.83	0.9925535
5.84	0.7664128	6.84	0.8350561	7.84	0.8943161	8.84	0.9464523	9.84	0.9929951
5.85	0.7671559	6.85	0.8356906	7.85	0.8948697	8.85	0.9469433	9.85	0.9934362
5.86	0.7678976	6.86	0.8363241	7.86	0.8954225	8.86	0.9474337	9.86	0.9938769
5.87	0.7686381	6.87	0.8369567	7.87

Write a Mixed Decimal

Mixed number: Mixed number is formed by associating a whole number with fraction numbers.

Decimal number: A number along decimal a point in it is called decimal number.

Mixed decimal number:  A number having a decimal point in its mixed format is mixed decimal number.

Example: 7.5 `3/6`.  Let us see how to write mixed decimal number.


Writing a mixed decimal:

Write a mixed decimal:

In the whole number part of mixed number the decimal point appears show that the number is mixed decimal.

It should be written as,

133.63 `1/5`

Conversion of mixed decimal into fraction:

Step 1: First see how many numbers present after the decimal point.

Step 2: Calculate number of 10’s equal to the counted digits after the decimal point.

Step 3: Put number of 10’s as denominator and the number is mixed decimal number.

Step 4: Convert the numerator mixed decimal number into fraction.

Step 5: Now simplify the terms by a common term.

Example problems for writing a mixed decimal:

Example: 1

Write the following mixed decimal number into fraction number.

12.5 2/3

Solution:

Given, mixed decimal number =12.5 2/3.

We have to add the 10 as the denominator and the mixed decimal number as numerator.

That is, `(125 2/3)/10`

= `(377/3)/10`

= `377/30`

=12`17/30`

Example: 2

Write the following mixed decimal number into fraction number.

5.22 1/2

Solution:

Given, mixed decimal number =5.22 1/2.

We have to add the 100 as the denominator and the mixed decimal number as numerator.

That is, `(522 1/2)/100`

= `(1045/2)/100`

= `1045/200`

=`209/20`

=1`9/20` .

Example: 3

Write the following mixed decimal number into fraction number.

15.123 1/5

Solution:

Given, mixed decimal number =15.123 1/5.

We have to add the 1000 as the denominator and the mixed decimal number as numerator.

That is, `(15123 1/5)/1000`

= `(75616/5)/1000`

= `75616/5000`

= 25 `616/5000`

= 25 `77/625` .

Answer: 25 `77/625` .

Practice problems for write a mixed decimal number:

Problem: 1

Write the following mixed decimal into mixed number.

5.9 3/4

Answer: 5`39/40`.

Problem: 2

Write the following mixed decimal into mixed number.

5.633 1/2 .

Answer: 5`1267/2000`.

Hypothesis Testing Variance

Hypothesis testing is the use of statistics  found in the probability that a specified hypothesis is correct. Hypothesis is specified as declaration which may or may not be accurate. In statistics two hypothesis testing are used. They are null hypothesis and alternative hypothesis. These two hypothesis tested are opposed to each other. In statistics the significance level is symbolized through alpha. Let us see about the probability of hypothesis testing variance.

Hypothesis testing variance

There are five constituent to either statistical test:

Null Hypothesis

Alternate Hypothesis

Test Statistic

Level of significance

Conclusion

Hypothesis testing variance

Consider the population is standard, we are able to test the variance of the method using the chi-square distribution through (n – 1) degrees of freedom.

To test a variance or standard deviation of a population to be exact normally distributed, we can utilize the χ2–test.

The χ2- test for a variance or standard deviation is not as robust as the samples for the population mean otherwise the population proportion.

Therefore it is necessary to while performing the χ2–test used for a variance that the population is usually circulated. The results can be deceptive if the population is not standard.

Examples for hypothesis testing variance

χ2- test for population variance

In a sample of size 16 drawn from a normal population standard deviation is 4 can you say that population standard deviation is 5.

Solution

Null hypothesis:

H0: The population standard deviation is 5

Test statistic:

χ2   =`(ns^2)/sigma^2` ~ χ2n-1

Level of significance:

α= 0.05 at 5% level of χ2  table values for 16 degrees of freedom is 24.996

Calculation:

n = 16, s = 4, σ = 5

χ2   =`(16xx16)/25 = 256/25`

χ2   = 10.24

Calculated value = 10.24

Table value = 24.996

Calculated value < Table value

Therefore the null hypothesis is accepted.

The population standard deviation is 5

Result

The population standard deviation is 5

Whole Set Fraction

This article is about whole set fraction. Whole set fraction is nothing but it is about the set of fractions. A fraction is a number that can be represented by an ordered pair of whole numbers `a/b` where `b!= 0`. Here a is represented as numerator and b as denominator. The tutors in tutor vista are always ready to help the students in any topics like whole set fraction. The online tutors help students in online. Tutor vista is the best tutoring website where many students follow this. Below we can see about the whole set of fraction.

whole set fractions

In set notation, the set of fractions is

F ={ `a/b` where a and b are whole numbers,  b` !=` 0 }

Two fractions that represent the same relative amount are termed to be equivalent fractions.

Proper Fraction:

When the numerator is less than the denominator then those fractions are called as Proper fraction.

Example: `2/3 `

Improper fraction:

When the numerator is greater than the denominator then this fraction is called as Improper fraction

Example: `7/5`

All the integers are simply a improper fraction

Example

3 is nothing but `3 / 1` which is an improper fraction

Mixed fraction:

Mixed fraction is a whole number with proper fraction

Example:  2 `1/3`

whole set fractions

Example 1:

Add `5/3`  + ` 8/3`

Here both the denominator is same

Add numerator alone.

`(5+8)/3`

`13/3`

Adding improper fraction with different denominator

Example 2:

Add `5/2`  + `4/3`

Here find LCM and solve to make the denominator equal

LCM of 2 and 3 is 6

`(5xx3)/ (2xx3) = 15/6`

`(4xx2)/ (3xx2) = 8/6`

So `(15+8)/6 = 23/6`

whole set fractions

Example 3: Add `3/11` +`6/11`

Solution

Here the denominators are same. So just add the numerator alone

` (3+6)/11`

`9/11` is the answer.

Example 4: Subtract `8/9` – `4/9`

Solution

Here the denominators are same. So just subtract the numerator alone.

`(8 - 4)/9`

`4/9`

Example 5: Multiply` 3/5` x `3/ 4 `

Solution

`3 / 5 ` x `3 / 4`

`(3xx3)/(5xx4) `

= `9 / 20 `

Addition Angle Formulas

Addition angle formula is based on trigonometric functions. We are having  the addition angle formulas to find the value of  the trigonometric equations and values for the trigonometric angles.
                         Cos (A+B) = Cos A Cos B - Sin A Sin B
                         Cos (A-B) = Cos A Cos B + Sin A Sin B
                         Sin (A+B) = Sin A Cos B + Cos A Sin B
                         Sin (A-B) = Sin A Cos B - Cos A Sin B
                         Tan (A+B) = `(Tan A + Tan B)/(1 - Tan A Tan B)`
                         Tan (A-B) =  `(Tan A - Tan B)/(1 + Tan A Tan B)`
                         Here we will some problems based on addition angle formulas.

Addition Angle Problems:

Problem 1:
         Solve the following trigonometric function using Addition angle formula Sin 75^o
Solution:
            Sin 75^o
                 We can write sin 75^o as Sin (45^o + 30^o)
                              We have the formula for Sin (A+B) = Sin A Cos B + Cos A Sin B
                              Where A = 45^o and  B = 30^o
                              Sin 45^o = Cos 45 = `(1)/(sqrt(2))`
                              Sin 30 = `(1)/(2)`    Cos 30 = `(sqrt(3))/(2)`
                              Sin (45^o + 30^o) = Sin 45^o Cos 30^o + Cos 45^o Sin 30^o
                                                          =  `(1)/(sqrt(2))` `(sqrt(3))/(2)` `(1)/(sqrt(2))` `(1)/(2)`
                                                          = `(sqrt(3))/(2sqrt(2))` + `(1)/(2sqrt(2))`
                                                          = `(sqrt(3)+1)/(2sqrt(2))`
Problem 2:
       Solve the following trigonometric function using Addition angle formula Cos 135^o
Solution:
               Cos 135^o
                We can write Cos 135^o as Sin (90^o + 45^o)
                              We have the formula for Cos (A+B) = Cos A Cos B - Sin A Sin B
                              Where A = 90^o and  B = 45^o
                              Sin 45^o = Cos 45 = `(1)/(sqrt(2))`
                              Sin 90^o = 1  Cos 90^o = 0
                              Cos (90^o + 45^o) = Cos 90^o Cos 45^o - Sin 90^o Sin 45^o
                                                      = 0 . `(1)/(sqrt(2))` - 1 . `(1)/(sqrt(2))`
                                                      = 0 - `(1)/(sqrt(2))`
                                                      = - `(1)/(sqrt(2))`

Problem 3:

       Solve the following trigonometric function using Addition angle formula Tan 135^o
Solution:
               Tan 135^o
               We can write Tan 135^o as Tan (180^o - 45^o)
                              We have the formula Tan (A - B) = `(Tan A - Tan B)/(1 + Tan A Tan B)`
                              Where A = 180^o and  B = 45^o
                              Tan 45^o = 1 Tan 180^o = 0
                              Tan 135^o = `(Tan 180^o - Tan 45^o)/(1 + Tan 180^o Tan 45^o)`
                              Tan 135^o = `"(0 - 1)/(1 + (0) (1)) `
                              Tan 135^o = `(- 1)/(1)`
                              Tan 135^o = -1