What is tg 45 degrees. Values of trigonometric functions at α = 45 °
Note: see also table of values of trigonometric functions of other angles.
Sine, cosine, tangent of an angle 45 degrees (sin 45, cos 45, tg 45)
Tabular values of sine 45, cosine 45 and tangent 45 degrees are indicated. Further in the text follows an explanation of the method and the correctness of calculating these values for an arbitrary right triangle.
45 degrees is π / 4 radians... The formulas for cosine, sine, and pi / 4 radians are shown below (although they are the same).
That is, for example, tg π / 4 = tg 45 degrees
VALUES OF TRIGONOMETRIC FUNCTIONS AT α = 45 °
How to calculate values yourself sin cos tg 45 degrees?
Let us construct and consider a right-angled triangle ABC whose angle ∠ B = 45 °. Based on the ratio of its sides, we calculate the values trigonometric functions in a right-angled triangle for an angle of 45 degrees. Since the triangle is rectangular, the values of the sine, cosine and tangent functions will be equal to the ratio of its respective sides.
Since the value of the sine, cosine and tangent functions depend exclusively on the degree measure of the angle (or the value expressed in radians), the relations we found will be the values of the sine function 45, cosine 45 and tangent 45 degrees.
According to the properties of a right-angled triangle, the angle C is a straight line and is equal to 90 degrees. We originally built Angle B with a 45 degree measure. Find the value of the angle A. Since the sum of the angles of a triangle is 180 degrees, then
∠
A + ∠
B + ∠
C = 180 °
Angle C is straight and is equal to 90 degrees, angle B was originally defined as 45 degrees, thus:
∠
A = 180 ° - ∠
WITH - ∠
B = 180 ° - 90 ° - 45 ° = 45 °
Since this triangle has two angles equal to each other, then triangle ABC - rectangular, and, at the same time, isosceles, in which both legs are equal: AC = BC.
Suppose that the length of the sides is equal to a certain number AC = BC = a. Knowing the length of the legs, we calculate the length of the hypotenuse.
By the Pythagorean theorem: AB 2 = AC 2 + BC 2
We replace the lengths AC and BC with the variable a, then we get:
AB 2 = a 2 + a 2 = 2a 2,
then AB = a √ 2.
As a result we expressed the lengths of all sides a right-angled triangle with an angle of 45 degrees through the variable a.
According to the properties of trigonometric functions in a right triangle the ratio of the corresponding sides of the triangle will be equal to the value of the corresponding functions... Thus, for an angle α = 45 degrees:
sin α = BC / AB(according to the definition of sine for a right-angled triangle, this is the ratio of the opposite leg to the hypotenuse, BC is the leg, AB is the hypotenuse)
cos α = AC / AB(according to the definition of cosine, this is the ratio of the adjacent leg to the hypotenuse, AC is the leg, AB is the hypotenuse)
tg α = BC / AC(similarly, the tangent for the angle α will be equal to the ratio of the opposite leg to the adjacent leg)
Instead of the designations of the sides, we substitute the values of their lengths through the variable a.
Based on this (see the table of values sin 45, cos 45, tg 45) we get:
Table values sin 45, cos 45, tg 45(that is, the value sine 45, cosine 45 and tangent 45 degrees can be calculated as the ratio of the corresponding sides of a given triangle), substitute the calculated above values of the lengths of the sides into the formulas and get the result in the picture below.
Tabular values: sine 45, cosine 45 and tangent 45 degrees
Thus:
- tangent 45 degrees is equal to one
- sine of 45 degrees is equal to cosine of 45 degrees and is equal to the root of two in half (same as one divided by the root of two)
As you can see from the calculations above, for calculating the values of the corresponding trigonometric function, it is not the lengths of the sides of the triangle that are important, but their ratio, which is always the same for the same angles, regardless of the size of a particular triangle.
Sine, cosine and tangent of an angle π / 4 radians
In the problems proposed for solving in high school and on the ZNO / USE, instead of the degree measure of the angle, an indication of its value, measured in radians, is often found. The measure of the angle, expressed in radians, is based on pi, which expresses the dependence of the circumference of a circle on its diameter.
For ease of understanding, I recommend remembering simple principle of converting degrees to radians... The diameter of the circle spans a 180-degree arc. So pi radians would be 180 degrees. From where it is easy to convert any degree measure of an angle into radians and vice versa.
Let's take into account that 45 degree angle expressed in radians, is equal to (180/45 = 4) π / 4 (pi by four). Therefore, the values we found are correct for the same degree measure of the angle, expressed in radians:
- tangent π / 4(pi by four) is equal to one
- sine π / 4(pi by four) degrees is cosine π / 4 degrees and is equal to the root of two in half
Tables of values of sines (sin), cosines (cos), tangents (tg), cotangents (ctg) is a powerful and useful tool that helps to solve many problems, both theoretical and applied. In this article, we provide a table of the main trigonometric functions (sines, cosines, tangents and cotangents) for angles 0, 30, 45, 60, 90, ..., 360 degrees (0, π 6, π 3, π 2,... ., 2 π radians). Separate Bradis tables for sines and cosines, tangents and cotangents will also be shown, with an explanation of how to use them to find the values of basic trigonometric functions.
Table of basic trigonometric functions for angles 0, 30, 45, 60, 90, ..., 360 degrees
Based on the definitions of sine, cosine, tangent and cotangent, you can find the values of these functions for angles of 0 and 90 degrees
sin 0 = 0, cos 0 = 1, t g 0 = 0, the cotangent of zero is not defined,
sin 90 ° = 1, cos 90 ° = 0, with t g 90 ° = 0, the tangent of the degree is not defined.
The values of sines, cosines, tangents and cotangents in the geometry course are defined as the aspect ratios of a right-angled triangle whose angles are 30, 60 and 90 degrees, and also 45, 45 and 90 degrees.
Determination of trigonometric functions for an acute angle in a right triangle
Sinus- the ratio of the opposite leg to the hypotenuse.
Cosine- the ratio of the adjacent leg to the hypotenuse.
Tangent- the ratio of the opposite leg to the adjacent one.
Cotangent- the ratio of the adjacent leg to the opposite one.
In accordance with the definitions, the values of the functions are found:
sin 30 ° = 1 2, cos 30 ° = 3 2, tg 30 ° = 3 3, ctg 30 ° = 3, sin 45 ° = 2 2, cos 45 ° = 2 2, tg 45 ° = 1, ctg 45 ° = 1, sin 60 ° = 3 2, cos 45 ° = 1 2, tg 45 ° = 3, ctg 45 ° = 3 3.
Let's summarize these values in a table and call it the table of the basic values of sine, cosine, tangent and cotangent.
α ° | 0 | 30 | 45 | 60 | 90 |
sin α | 0 | 1 2 | 2 2 | 3 2 | 1 |
cos α | 1 | 3 2 | 2 2 | 1 2 | 0 |
t g α | 0 | 3 3 | 1 | 3 | indefined |
c t g α | indefined | 3 | 1 | 3 3 | 0 |
α, r a d i an n | 0 | π 6 | π 4 | π 3 | π 2 |
One of the important properties of trigonometric functions is periodicity. Based on this property, this table can be extended using cast formulas. Below we present an expanded table of values of the main trigonometric functions for angles 0, 30, 60, ..., 120, 135, 150, 180, ..., 360 degrees (0, π 6, π 3, π 2, ..., 2 π radians).
α ° | 0 | 30 | 45 | 60 | 90 | 120 | 135 | 150 | 180 | 210 | 225 | 240 | 270 | 300 | 315 | 330 | 360 |
sin α | 0 | 1 2 | 2 2 | 3 2 | 1 | 3 2 | 2 2 | 1 2 | 0 | - 1 2 | - 2 2 | - 3 2 | - 1 | - 3 2 | - 2 2 | - 1 2 | 0 |
cos α | 1 | 3 2 | 2 2 | 1 2 | 0 | - 1 2 | - 2 2 | - 3 2 | - 1 | - 3 2 | - 2 2 | - 1 2 | 0 | 1 2 | 2 2 | 3 2 | 1 |
t g α | 0 | 3 3 | 1 | 3 | - | - 1 | - 3 3 | 0 | 0 | 3 3 | 1 | 3 | - | - 3 | - 1 | 0 | |
c t g α | - | 3 | 1 | 3 3 | 0 | - 3 3 | - 1 | - 3 | - | 3 | 1 | 3 3 | 0 | - 3 3 | - 1 | - 3 | - |
α, r a d i an n | 0 | π 6 | π 4 | π 3 | π 2 | 2 π 3 | 3 π 4 | 5 π 6 | π | 7 π 6 | 5 π 4 | 4 π 3 | 3 π 2 | 5 π 3 | 7 π 4 | 11 π 6 | 2 π |
The periodicity of sine, cosine, tangent and cotangent allows you to expand this table to arbitrarily large values of the angles. The values collected in the table are used most often in solving problems, so it is recommended to memorize them.
How to use the table of basic values of trigonometric functions
The principle of using the table of values of sines, cosines, tangents and cotangents is intuitive. The intersection of the row and column gives the function value for that particular corner.
Example. How to use the table of sines, cosines, tangents and cotangents
You need to find out what is sin 7 π 6
Find a column in the table, the value of the last cell of which is 7 π 6 radians - the same as 210 degrees. Then we select the term of the table in which the values of the sines are presented. At the intersection of the row and column, we find the desired value:
sin 7 π 6 = - 1 2
Bradis tables
The Bradis table allows you to calculate the value of sine, cosine, tangent or cotangent with an accuracy of 4 decimal places without the use of computer technology. This is a kind of replacement for an engineering calculator.
reference
Vladimir Modestovich Bradis (1890 - 1975) - Soviet mathematician-teacher, since 1954 a corresponding member of the USSR Academy of Pedagogical Sciences. Bradis' tables of four-digit logarithms and natural trigonometric values were first published in 1921.
First, we give the Bradis table for sines and cosines. It allows you to accurately calculate the approximate values of these functions for angles containing an integer number of degrees and minutes. The leftmost column of the table shows degrees, and the top row shows minutes. Note that all angles in the Bradis table are multiples of six minutes.
Bradis table for sines and cosines
sin | 0" | 6" | 12" | 18" | 24" | 30" | 36" | 42" | 48" | 54" | 60" | cos | 1" | 2" | 3" |
0.0000 | 90 ° | ||||||||||||||
0° | 0.0000 | 0017 | 0035 | 0052 | 0070 | 0087 | 0105 | 0122 | 0140 | 0157 | 0175 | 89 ° | 3 | 6 | 9 |
1 ° | 0175 | 0192 | 0209 | 0227 | 0244 | 0262 | 0279 | 0297 | 0314 | 0332 | 0349 | 88 ° | 3 | 6 | 9 |
2 ° | 0349 | 0366 | 0384 | 0401 | 0419 | 0436 | 0454 | 0471 | 0488 | 0506 | 0523 | 87 ° | 3 | 6 | 9 |
3 ° | 0523 | 0541 | 0558 | 0576 | 0593 | 0610 | 0628 | 0645 | 0663 | 0680 | 0698 | 86 ° | 3 | 6 | 9 |
4 ° | 0698 | 0715 | 0732 | 0750 | 0767 | 0785 | 0802 | 0819 | 0837 | 0854 | 0.0872 | 85 ° | 3 | 6 | 9 |
5 ° | 0.0872 | 0889 | 0906 | 0924 | 0941 | 0958 | 0976 | 0993 | 1011 | 1028 | 1045 | 84 ° | 3 | 6 | 9 |
6 ° | 1045 | 1063 | 1080 | 1097 | 1115 | 1132 | 1149 | 1167 | 1184 | 1201 | 1219 | 83 ° | 3 | 6 | 9 |
7 ° | 1219 | 1236 | 1253 | 1271 | 1288 | 1305 | 1323 | 1340 | 1357 | 1374 | 1392 | 82 ° | 3 | 6 | 9 |
8 ° | 1392 | 1409 | 1426 | 1444 | 1461 | 1478 | 1495 | 1513 | 1530 | 1547 | 1564 | 81 ° | 3 | 6 | 9 |
9 ° | 1564 | 1582 | 1599 | 1616 | 1633 | 1650 | 1668 | 1685 | 1702 | 1719 | 0.1736 | 80 ° | 3 | 6 | 9 |
10 ° | 0.1736 | 1754 | 1771 | 1788 | 1805 | 1822 | 1840 | 1857 | 1874 | 1891 | 1908 | 79 ° | 3 | 6 | 9 |
11 ° | 1908 | 1925 | 1942 | 1959 | 1977 | 1994 | 2011 | 2028 | 2045 | 2062 | 2079 | 78 ° | 3 | 6 | 9 |
12 ° | 2079 | 2096 | 2113 | 2130 | 2147 | 2164 | 2181 | 2198 | 2215 | 2233 | 2250 | 77 ° | 3 | 6 | 9 |
13 ° | 2250 | 2267 | 2284 | 2300 | 2317 | 2334 | 2351 | 2368 | 2385 | 2402 | 2419 | 76 ° | 3 | 6 | 8 |
14 ° | 2419 | 2436 | 2453 | 2470 | 2487 | 2504 | 2521 | 2538 | 2554 | 2571 | 0.2588 | 75 ° | 3 | 6 | 8 |
15 ° | 0.2588 | 2605 | 2622 | 2639 | 2656 | 2672 | 2689 | 2706 | 2723 | 2740 | 2756 | 74 ° | 3 | 6 | 8 |
16 ° | 2756 | 2773 | 2790 | 2807 | 2823 | 2840 | 2857 | 2874 | 2890 | 2907 | 2924 | 73 ° | 3 | 6 | 8 |
17 ° | 2924 | 2940 | 2957 | 2974 | 2990 | 3007 | 3024 | 3040 | 3057 | 3074 | 3090 | 72 ° | 3 | 6 | 8 |
18 ° | 3090 | 3107 | 3123 | 3140 | 3156 | 3173 | 3190 | 3206 | 3223 | 3239 | 3256 | 71 ° | 3 | 6 | 8 |
19 ° | 3256 | 3272 | 3289 | 3305 | 3322 | 3338 | 3355 | 3371 | 3387 | 3404 | 0.3420 | 70 ° | 3 | 5 | 8 |
20 ° | 0.3420 | 3437 | 3453 | 3469 | 3486 | 3502 | 3518 | 3535 | 3551 | 3567 | 3584 | 69 ° | 3 | 5 | 8 |
21 ° | 3584 | 3600 | 3616 | 3633 | 3649 | 3665 | 3681 | 3697 | 3714 | 3730 | 3746 | 68 ° | 3 | 5 | 8 |
22 ° | 3746 | 3762 | 3778 | 3795 | 3811 | 3827 | 3843 | 3859 | 3875 | 3891 | 3907 | 67 ° | 3 | 5 | 8 |
23 ° | 3907 | 3923 | 3939 | 3955 | 3971 | 3987 | 4003 | 4019 | 4035 | 4051 | 4067 | 66 ° | 3 | 5 | 8 |
24 ° | 4067 | 4083 | 4099 | 4115 | 4131 | 4147 | 4163 | 4179 | 4195 | 4210 | 0.4226 | 65 ° | 3 | 5 | 8 |
25 ° | 0.4226 | 4242 | 4258 | 4274 | 4289 | 4305 | 4321 | 4337 | 4352 | 4368 | 4384 | 64 ° | 3 | 5 | 8 |
26 ° | 4384 | 4399 | 4415 | 4431 | 4446 | 4462 | 4478 | 4493 | 4509 | 4524 | 4540 | 63 ° | 3 | 5 | 8 |
27 ° | 4540 | 4555 | 4571 | 4586 | 4602 | 4617 | 4633 | 4648 | 4664 | 4679 | 4695 | 62 ° | 3 | 5 | 8 |
28 ° | 4695 | 4710 | 4726 | 4741 | 4756 | 4772 | 4787 | 4802 | 4818 | 4833 | 4848 | 61 ° | 3 | 5 | 8 |
29 ° | 4848 | 4863 | 4879 | 4894 | 4909 | 4924 | 4939 | 4955 | 4970 | 4985 | 0.5000 | 60 ° | 3 | 5 | 8 |
30 ° | 0.5000 | 5015 | 5030 | 5045 | 5060 | 5075 | 5090 | 5105 | 5120 | 5135 | 5150 | 59 ° | 3 | 5 | 8 |
31 ° | 5150 | 5165 | 5180 | 5195 | 5210 | 5225 | 5240 | 5255 | 5270 | 5284 | 5299 | 58 ° | 2 | 5 | 7 |
32 ° | 5299 | 5314 | 5329 | 5344 | 5358 | 5373 | 5388 | 5402 | 5417 | 5432 | 5446 | 57 ° | 2 | 5 | 7 |
33 ° | 5446 | 5461 | 5476 | 5490 | 5505 | 5519 | 5534 | 5548 | 5563 | 5577 | 5592 | 56 ° | 2 | 5 | 7 |
34 ° | 5592 | 5606 | 5621 | 5635 | 5650 | 5664 | 5678 | 5693 | 5707 | 5721 | 0.5736 | 55 ° | 2 | 5 | 7 |
35 ° | 0.5736 | 5750 | 5764 | 5779 | 5793 | 5807 | 5821 | 5835 | 5850 | 5864 | 0.5878 | 54 ° | 2 | 5 | 7 |
36 ° | 5878 | 5892 | 5906 | 5920 | 5934 | 5948 | 5962 | 5976 | 5990 | 6004 | 6018 | 53 ° | 2 | 5 | 7 |
37 ° | 6018 | 6032 | 6046 | 6060 | 6074 | 6088 | 6101 | 6115 | 6129 | 6143 | 6157 | 52 ° | 2 | 5 | 7 |
38 ° | 6157 | 6170 | 6184 | 6198 | 6211 | 6225 | 6239 | 6252 | 6266 | 6280 | 6293 | 51 ° | 2 | 5 | 7 |
39 ° | 6293 | 6307 | 6320 | 6334 | 6347 | 6361 | 6374 | 6388 | 6401 | 6414 | 0.6428 | 50 ° | 2 | 4 | 7 |
40 ° | 0.6428 | 6441 | 6455 | 6468 | 6481 | 6494 | 6508 | 6521 | 6534 | 6547 | 6561 | 49 ° | 2 | 4 | 7 |
41 ° | 6561 | 6574 | 6587 | 6600 | 6613 | 6626 | 6639 | 6652 | 6665 | 6678 | 6691 | 48 ° | 2 | 4 | 7 |
42 ° | 6691 | 6704 | 6717 | 6730 | 6743 | 6756 | 6769 | 6782 | 6794 | 6807 | 6820 | 47 ° | 2 | 4 | 6 |
43 ° | 6820 | 6833 | 6845 | 6858 | 6871 | 6884 | 6896 | 8909 | 6921 | 6934 | 6947 | 46 ° | 2 | 4 | 6 |
44 ° | 6947 | 6959 | 6972 | 6984 | 6997 | 7009 | 7022 | 7034 | 7046 | 7059 | 0.7071 | 45 ° | 2 | 4 | 6 |
45 ° | 0.7071 | 7083 | 7096 | 7108 | 7120 | 7133 | 7145 | 7157 | 7169 | 7181 | 7193 | 44 ° | 2 | 4 | 6 |
46 ° | 7193 | 7206 | 7218 | 7230 | 7242 | 7254 | 7266 | 7278 | 7290 | 7302 | 7314 | 43 ° | 2 | 4 | 6 |
47 ° | 7314 | 7325 | 7337 | 7349 | 7361 | 7373 | 7385 | 7396 | 7408 | 7420 | 7431 | 42 ° | 2 | 4 | 6 |
48 ° | 7431 | 7443 | 7455 | 7466 | 7478 | 7490 | 7501 | 7513 | 7524 | 7536 | 7547 | 41 ° | 2 | 4 | 6 |
49 ° | 7547 | 7559 | 7570 | 7581 | 7593 | 7604 | 7615 | 7627 | 7638 | 7649 | 0.7660 | 40 ° | 2 | 4 | 6 |
50 ° | 0.7660 | 7672 | 7683 | 7694 | 7705 | 7716 | 7727 | 7738 | 7749 | 7760 | 7771 | 39 ° | 2 | 4 | 6 |
51 ° | 7771 | 7782 | 7793 | 7804 | 7815 | 7826 | 7837 | 7848 | 7859 | 7869 | 7880 | 38 ° | 2 | 4 | 5 |
52 ° | 7880 | 7891 | 7902 | 7912 | 7923 | 7934 | 7944 | 7955 | 7965 | 7976 | 7986 | 37 ° | 2 | 4 | 5 |
53 ° | 7986 | 7997 | 8007 | 8018 | 8028 | 8039 | 8049 | 8059 | 8070 | 8080 | 8090 | 36 ° | 2 | 3 | 5 |
54 ° | 8090 | 8100 | 8111 | 8121 | 8131 | 8141 | 8151 | 8161 | 8171 | 8181 | 0.8192 | 35 ° | 2 | 3 | 5 |
55 ° | 0.8192 | 8202 | 8211 | 8221 | 8231 | 8241 | 8251 | 8261 | 8271 | 8281 | 8290 | 34 ° | 2 | 3 | 5 |
56 ° | 8290 | 8300 | 8310 | 8320 | 8329 | 8339 | 8348 | 8358 | 8368 | 8377 | 8387 | 33 ° | 2 | 3 | 5 |
57 ° | 8387 | 8396 | 8406 | 8415 | 8425 | 8434 | 8443 | 8453 | 8462 | 8471 | 8480 | 32 ° | 2 | 3 | 5 |
58 ° | 8480 | 8490 | 8499 | 8508 | 8517 | 8526 | 8536 | 8545 | 8554 | 8563 | 8572 | 31 ° | 2 | 3 | 5 |
59 ° | 8572 | 8581 | 8590 | 8599 | 8607 | 8616 | 8625 | 8634 | 8643 | 8652 | 0.8660 | 30 ° | 1 | 3 | 4 |
60 ° | 0.8660 | 8669 | 8678 | 8686 | 8695 | 8704 | 8712 | 8721 | 8729 | 8738 | 8746 | 29 ° | 1 | 3 | 4 |
61 ° | 8746 | 8755 | 8763 | 8771 | 8780 | 8788 | 8796 | 8805 | 8813 | 8821 | 8829 | 28 ° | 1 | 3 | 4 |
62 ° | 8829 | 8838 | 8846 | 8854 | 8862 | 8870 | 8878 | 8886 | 8894 | 8902 | 8910 | 27 ° | 1 | 3 | 4 |
63 ° | 8910 | 8918 | 8926 | 8934 | 8942 | 8949 | 8957 | 8965 | 8973 | 8980 | 8988 | 26 ° | 1 | 3 | 4 |
64 ° | 8988 | 8996 | 9003 | 9011 | 9018 | 9026 | 9033 | 9041 | 9048 | 9056 | 0.9063 | 25 ° | 1 | 3 | 4 |
65 ° | 0.9063 | 9070 | 9078 | 9085 | 9092 | 9100 | 9107 | 9114 | 9121 | 9128 | 9135 | 24 ° | 1 | 2 | 4 |
66 ° | 9135 | 9143 | 9150 | 9157 | 9164 | 9171 | 9178 | 9184 | 9191 | 9198 | 9205 | 23 ° | 1 | 2 | 3 |
67 ° | 9205 | 9212 | 9219 | 9225 | 9232 | 9239 | 9245 | 9252 | 9259 | 9256 | 9272 | 22 ° | 1 | 2 | 3 |
68 ° | 9272 | 9278 | 9285 | 9291 | 9298 | 9304 | 9311 | 9317 | 9323 | 9330 | 9336 | 21 ° | 1 | 2 | 3 |
69 ° | 9336 | 9342 | 9348 | 9354 | 9361 | 9367 | 9373 | 9379 | 9383 | 9391 | 0.9397 | 20 ° | 1 | 2 | 3 |
70 ° | 9397 | 9403 | 9409 | 9415 | 9421 | 9426 | 9432 | 9438 | 9444 | 9449 | 0.9455 | 19 ° | 1 | 2 | 3 |
71 ° | 9455 | 9461 | 9466 | 9472 | 9478 | 9483 | 9489 | 9494 | 9500 | 9505 | 9511 | 18 ° | 1 | 2 | 3 |
72 ° | 9511 | 9516 | 9521 | 9527 | 9532 | 9537 | 9542 | 9548 | 9553 | 9558 | 9563 | 17 ° | 1 | 2 | 3 |
73 ° | 9563 | 9568 | 9573 | 9578 | 9583 | 9588 | 9593 | 9598 | 9603 | 9608 | 9613 | 16 ° | 1 | 2 | 2 |
74 ° | 9613 | 9617 | 9622 | 9627 | 9632 | 9636 | 9641 | 9646 | 9650 | 9655 | 0.9659 | 15 ° | 1 | 2 | 2 |
75 ° | 9659 | 9664 | 9668 | 9673 | 9677 | 9681 | 9686 | 9690 | 9694 | 9699 | 9703 | 14 ° | 1 | 1 | 2 |
76 ° | 9703 | 9707 | 9711 | 9715 | 9720 | 9724 | 9728 | 9732 | 9736 | 9740 | 9744 | 13 ° | 1 | 1 | 2 |
77 ° | 9744 | 9748 | 9751 | 9755 | 9759 | 9763 | 9767 | 9770 | 9774 | 9778 | 9781 | 12 ° | 1 | 1 | 2 |
78 ° | 9781 | 9785 | 9789 | 9792 | 9796 | 9799 | 9803 | 9806 | 9810 | 9813 | 9816 | 11 ° | 1 | 1 | 2 |
79 ° | 9816 | 9820 | 9823 | 9826 | 9829 | 9833 | 9836 | 9839 | 9842 | 9845 | 0.9848 | 10 ° | 1 | 1 | 2 |
80 ° | 0.9848 | 9851 | 9854 | 9857 | 9860 | 9863 | 9866 | 9869 | 9871 | 9874 | 9877 | 9 ° | 0 | 1 | 1 |
81 ° | 9877 | 9880 | 9882 | 9885 | 9888 | 9890 | 9893 | 9895 | 9898 | 9900 | 9903 | 8 ° | 0 | 1 | 1 |
82 ° | 9903 | 9905 | 9907 | 9910 | 9912 | 9914 | 9917 | 9919 | 9921 | 9923 | 9925 | 7 ° | 0 | 1 | 1 |
83 ° | 9925 | 9928 | 9930 | 9932 | 9934 | 9936 | 9938 | 9940 | 9942 | 9943 | 9945 | 6 ° | 0 | 1 | 1 |
84 ° | 9945 | 9947 | 9949 | 9951 | 9952 | 9954 | 9956 | 9957 | 9959 | 9960 | 9962 | 5 ° | 0 | 1 | 1 |
85 ° | 9962 | 9963 | 9965 | 9966 | 9968 | 9969 | 9971 | 9972 | 9973 | 9974 | 9976 | 4 ° | 0 | 0 | 1 |
86 ° | 9976 | 9977 | 9978 | 9979 | 9980 | 9981 | 9982 | 9983 | 9984 | 9985 | 9986 | 3 ° | 0 | 0 | 0 |
87 ° | 9986 | 9987 | 9988 | 9989 | 9990 | 9990 | 9991 | 9992 | 9993 | 9993 | 9994 | 2 ° | 0 | 0 | 0 |
88 ° | 9994 | 9995 | 9995 | 9996 | 9996 | 9997 | 9997 | 9997 | 9998 | 9998 | 0.9998 | 1 ° | 0 | 0 | 0 |
89 ° | 9998 | 9999 | 9999 | 9999 | 9999 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0° | 0 | 0 | 0 |
90 ° | 1.0000 | ||||||||||||||
sin | 60" | 54" | 48" | 42" | 36" | 30" | 24" | 18" | 12" | 6" | 0" | cos | 1" | 2" | 3" |
To find the values of sines and cosines of angles that are not presented in the table, it is necessary to use corrections.
Now we give the Bradis table for tangents and cotangents. It contains tangents of angles from 0 to 76 degrees, and cotangents of angles from 14 to 90 degrees.
Bradis table for tangent and cotangent
tg | 0" | 6" | 12" | 18" | 24" | 30" | 36" | 42" | 48" | 54" | 60" | ctg | 1" | 2" | 3" |
0 | 90 ° | ||||||||||||||
0° | 0,000 | 0017 | 0035 | 0052 | 0070 | 0087 | 0105 | 0122 | 0140 | 0157 | 0175 | 89 ° | 3 | 6 | 9 |
1 ° | 0175 | 0192 | 0209 | 0227 | 0244 | 0262 | 0279 | 0297 | 0314 | 0332 | 0349 | 88 ° | 3 | 6 | 9 |
2 ° | 0349 | 0367 | 0384 | 0402 | 0419 | 0437 | 0454 | 0472 | 0489 | 0507 | 0524 | 87 ° | 3 | 6 | 9 |
3 ° | 0524 | 0542 | 0559 | 0577 | 0594 | 0612 | 0629 | 0647 | 0664 | 0682 | 0699 | 86 ° | 3 | 6 | 9 |
4 ° | 0699 | 0717 | 0734 | 0752 | 0769 | 0787 | 0805 | 0822 | 0840 | 0857 | 0,0875 | 85 ° | 3 | 6 | 9 |
5 ° | 0,0875 | 0892 | 0910 | 0928 | 0945 | 0963 | 0981 | 0998 | 1016 | 1033 | 1051 | 84 ° | 3 | 6 | 9 |
6 ° | 1051 | 1069 | 1086 | 1104 | 1122 | 1139 | 1157 | 1175 | 1192 | 1210 | 1228 | 83 ° | 3 | 6 | 9 |
7 ° | 1228 | 1246 | 1263 | 1281 | 1299 | 1317 | 1334 | 1352 | 1370 | 1388 | 1405 | 82 ° | 3 | 6 | 9 |
8 ° | 1405 | 1423 | 1441 | 1459 | 1477 | 1495 | 1512 | 1530 | 1548 | 1566 | 1584 | 81 ° | 3 | 6 | 9 |
9 ° | 1584 | 1602 | 1620 | 1638 | 1655 | 1673 | 1691 | 1709 | 1727 | 1745 | 0,1763 | 80 ° | 3 | 6 | 9 |
10 ° | 0,1763 | 1781 | 1799 | 1817 | 1835 | 1853 | 1871 | 1890 | 1908 | 1926 | 1944 | 79 ° | 3 | 6 | 9 |
11 ° | 1944 | 1962 | 1980 | 1998 | 2016 | 2035 | 2053 | 2071 | 2089 | 2107 | 2126 | 78 ° | 3 | 6 | 9 |
12 ° | 2126 | 2144 | 2162 | 2180 | 2199 | 2217 | 2235 | 2254 | 2272 | 2290 | 2309 | 77 ° | 3 | 6 | 9 |
13 ° | 2309 | 2327 | 2345 | 2364 | 2382 | 2401 | 2419 | 2438 | 2456 | 2475 | 2493 | 76 ° | 3 | 6 | 9 |
14 ° | 2493 | 2512 | 2530 | 2549 | 2568 | 2586 | 2605 | 2623 | 2642 | 2661 | 0,2679 | 75 ° | 3 | 6 | 9 |
15 ° | 0,2679 | 2698 | 2717 | 2736 | 2754 | 2773 | 2792 | 2811 | 2830 | 2849 | 2867 | 74 ° | 3 | 6 | 9 |
16 ° | 2867 | 2886 | 2905 | 2924 | 2943 | 2962 | 2981 | 3000 | 3019 | 3038 | 3057 | 73 ° | 3 | 6 | 9 |
17 ° | 3057 | 3076 | 3096 | 3115 | 3134 | 3153 | 3172 | 3191 | 3211 | 3230 | 3249 | 72 ° | 3 | 6 | 10 |
18 ° | 3249 | 3269 | 3288 | 3307 | 3327 | 3346 | 3365 | 3385 | 3404 | 3424 | 3443 | 71 ° | 3 | 6 | 10 |
19 ° | 3443 | 3463 | 3482 | 3502 | 3522 | 3541 | 3561 | 3581 | 3600 | 3620 | 0,3640 | 70 ° | 3 | 7 | 10 |
20 ° | 0,3640 | 3659 | 3679 | 3699 | 3719 | 3739 | 3759 | 3779 | 3799 | 3819 | 3839 | 69 ° | 3 | 7 | 10 |
21 ° | 3839 | 3859 | 3879 | 3899 | 3919 | 3939 | 3959 | 3979 | 4000 | 4020 | 4040 | 68 ° | 3 | 7 | 10 |
22 ° | 4040 | 4061 | 4081 | 4101 | 4122 | 4142 | 4163 | 4183 | 4204 | 4224 | 4245 | 67 ° | 3 | 7 | 10 |
23 ° | 4245 | 4265 | 4286 | 4307 | 4327 | 4348 | 4369 | 4390 | 4411 | 4431 | 4452 | 66 ° | 3 | 7 | 10 |
24 ° | 4452 | 4473 | 4494 | 4515 | 4536 | 4557 | 4578 | 4599 | 4621 | 4642 | 0,4663 | 65 ° | 4 | 7 | 11 |
25 ° | 0,4663 | 4684 | 4706 | 4727 | 4748 | 4770 | 4791 | 4813 | 4834 | 4856 | 4877 | 64 ° | 4 | 7 | 11 |
26 ° | 4877 | 4899 | 4921 | 4942 | 4964 | 4986 | 5008 | 5029 | 5051 | 5073 | 5095 | 63 ° | 4 | 7 | 11 |
27 ° | 5095 | 5117 | 5139 | 5161 | 5184 | 5206 | 5228 | 5250 | 5272 | 5295 | 5317 | 62 ° | 4 | 7 | 11 |
28 ° | 5317 | 5340 | 5362 | 5384 | 5407 | 5430 | 5452 | 5475 | 5498 | 5520 | 5543 | 61 ° | 4 | 8 | 11 |
29 ° | 5543 | 5566 | 5589 | 5612 | 5635 | 5658 | 5681 | 5704 | 5727 | 5750 | 0,5774 | 60 ° | 4 | 8 | 12 |
30 ° | 0,5774 | 5797 | 5820 | 5844 | 5867 | 5890 | 5914 | 5938 | 5961 | 5985 | 6009 | 59 ° | 4 | 8 | 12 |
31 ° | 6009 | 6032 | 6056 | 6080 | 6104 | 6128 | 6152 | 6176 | 6200 | 6224 | 6249 | 58 ° | 4 | 8 | 12 |
32 ° | 6249 | 6273 | 6297 | 6322 | 6346 | 6371 | 6395 | 6420 | 6445 | 6469 | 6494 | 57 ° | 4 | 8 | 12 |
33 ° | 6494 | 6519 | 6544 | 6569 | 6594 | 6619 | 6644 | 6669 | 6694 | 6720 | 6745 | 56 ° | 4 | 8 | 13 |
34 ° | 6745 | 6771 | 6796 | 6822 | 6847 | 6873 | 6899 | 6924 | 6950 | 6976 | 0,7002 | 55 ° | 4 | 9 | 13 |
35 ° | 0,7002 | 7028 | 7054 | 7080 | 7107 | 7133 | 7159 | 7186 | 7212 | 7239 | 7265 | 54 ° | 4 | 8 | 13 |
36 ° | 7265 | 7292 | 7319 | 7346 | 7373 | 7400 | 7427 | 7454 | 7481 | 7508 | 7536 | 53 ° | 5 | 9 | 14 ° |
37 ° | 7536 | 7563 | 7590 | 7618 | 7646 | 7673 | 7701 | 7729 | 7757 | 7785 | 7813 | 52 ° | 5 | 9 | 14 |
38 ° | 7813 | 7841 | 7869 | 7898 | 7926 | 7954 | 7983 | 8012 | 8040 | 8069 | 8098 | 51 ° | 5 | 9 | 14 |
39 ° | 8098 | 8127 | 8156 | 8185 | 8214 | 8243 | 8273 | 8302 | 8332 | 8361 | 0,8391 | 50 ° | 5 | 10 | 15 |
40 ° | 0,8391 | 8421 | 8451 | 8481 | 8511 | 8541 | 8571 | 8601 | 8632 | 8662 | 0,8693 | 49 ° | 5 | 10 | 15 |
41 ° | 8693 | 8724 | 8754 | 8785 | 8816 | 8847 | 8878 | 8910 | 8941 | 8972 | 9004 | 48 ° | 5 | 10 | 16 |
42 ° | 9004 | 9036 | 9067 | 9099 | 9131 | 9163 | 9195 | 9228 | 9260 | 9293 | 9325 | 47 ° | 6 | 11 | 16 |
43 ° | 9325 | 9358 | 9391 | 9424 | 9457 | 9490 | 9523 | 9556 | 9590 | 9623 | 0,9657 | 46 ° | 6 | 11 | 17 |
44 ° | 9657 | 9691 | 9725 | 9759 | 9793 | 9827 | 9861 | 9896 | 9930 | 9965 | 1,0000 | 45 ° | 6 | 11 | 17 |
45 ° | 1,0000 | 0035 | 0070 | 0105 | 0141 | 0176 | 0212 | 0247 | 0283 | 0319 | 0355 | 44 ° | 6 | 12 | 18 |
46 ° | 0355 | 0392 | 0428 | 0464 | 0501 | 0538 | 0575 | 0612 | 0649 | 0686 | 0724 | 43 ° | 6 | 12 | 18 |
47 ° | 0724 | 0761 | 0799 | 0837 | 0875 | 0913 | 0951 | 0990 | 1028 | 1067 | 1106 | 42 ° | 6 | 13 | 19 |
48 ° | 1106 | 1145 | 1184 | 1224 | 1263 | 1303 | 1343 | 1383 | 1423 | 1463 | 1504 | 41 ° | 7 | 13 | 20 |
49 ° | 1504 | 1544 | 1585 | 1626 | 1667 | 1708 | 1750 | 1792 | 1833 | 1875 | 1,1918 | 40 ° | 7 | 14 | 21 |
50 ° | 1,1918 | 1960 | 2002 | 2045 | 2088 | 2131 | 2174 | 2218 | 2261 | 2305 | 2349 | 39 ° | 7 | 14 | 22 |
51 ° | 2349 | 2393 | 2437 | 2482 | 2527 | 2572 | 2617 | 2662 | 2708 | 2753 | 2799 | 38 ° | 8 | 15 | 23 |
52 ° | 2799 | 2846 | 2892 | 2938 | 2985 | 3032 | 3079 | 3127 | 3175 | 3222 | 3270 | 37 ° | 8 | 16 | 24 |
53 ° | 3270 | 3319 | 3367 | 3416 | 3465 | 3514 | 3564 | 3613 | 3663 | 3713 | 3764 | 36 ° | 8 | 16 | 25 |
54 ° | 3764 | 3814 | 3865 | 3916 | 3968 | 4019 | 4071 | 4124 | 4176 | 4229 | 1,4281 | 35 ° | 9 | 17 | 26 |
55 ° | 1,4281 | 4335 | 4388 | 4442 | 4496 | 4550 | 4605 | 4659 | 4715 | 4770 | 4826 | 34 ° | 9 | 18 | 27 |
56 ° | 4826 | 4882 | 4938 | 4994 | 5051 | 5108 | 5166 | 5224 | 5282 | 5340 | 5399 | 33 ° | 10 | 19 | 29 |
57 ° | 5399 | 5458 | 5517 | 5577 | 5637 | 5697 | 5757 | 5818 | 5880 | 5941 | 6003 | 32 ° | 10 | 20 | 30 |
58 ° | 6003 | 6066 | 6128 | 6191 | 6255 | 6319 | 6383 | 6447 | 6512 | 6577 | 6643 | 31 ° | 11 | 21 | 32 |
59 ° | 6643 | 6709 | 6775 | 6842 | 6909 | 6977 | 7045 | 7113 | 7182 | 7251 | 1,7321 | 30 ° | 11 | 23 | 34 |
60 ° | 1,732 | 1,739 | 1,746 | 1,753 | 1,760 | 1,767 | 1,775 | 1,782 | 1,789 | 1,797 | 1,804 | 29 ° | 1 | 2 | 4 |
61 ° | 1,804 | 1,811 | 1,819 | 1,827 | 1,834 | 1,842 | 1,849 | 1,857 | 1,865 | 1,873 | 1,881 | 28 ° | 1 | 3 | 4 |
62 ° | 1,881 | 1,889 | 1,897 | 1,905 | 1,913 | 1,921 | 1,929 | 1,937 | 1,946 | 1,954 | 1,963 | 27 ° | 1 | 3 | 4 |
63 ° | 1,963 | 1,971 | 1,980 | 1,988 | 1,997 | 2,006 | 2,014 | 2,023 | 2,032 | 2,041 | 2,05 | 26 ° | 1 | 3 | 4 |
64 ° | 2,050 | 2,059 | 2,069 | 2,078 | 2,087 | 2,097 | 2,106 | 2,116 | 2,125 | 2,135 | 2,145 | 25 ° | 2 | 3 | 5 |
65 ° | 2,145 | 2,154 | 2,164 | 2,174 | 2,184 | 2,194 | 2,204 | 2,215 | 2,225 | 2,236 | 2,246 | 24 ° | 2 | 3 | 5 |
66 ° | 2,246 | 2,257 | 2,267 | 2,278 | 2,289 | 2,3 | 2,311 | 2,322 | 2,333 | 2,344 | 2,356 | 23 ° | 2 | 4 | 5 |
67 ° | 2,356 | 2,367 | 2,379 | 2,391 | 2,402 | 2,414 | 2,426 | 2,438 | 2,450 | 2,463 | 2,475 | 22 ° | 2 | 4 | 6 |
68 ° | 2,475 | 2,488 | 2,5 | 2,513 | 2,526 | 2,539 | 2,552 | 2,565 | 2,578 | 2,592 | 2,605 | 21 ° | 2 | 4 | 6 |
69 ° | 2,605 | 2,619 | 2,633 | 2,646 | 2,66 | 2,675 | 2,689 | 2,703 | 2,718 | 2,733 | 2,747 | 20 ° | 2 | 5 | 7 |
70 ° | 2,747 | 2,762 | 2,778 | 2,793 | 2,808 | 2,824 | 2,840 | 2,856 | 2,872 | 2,888 | 2,904 | 19 ° | 3 | 5 | 8 |
71 ° | 2,904 | 2,921 | 2,937 | 2,954 | 2,971 | 2,989 | 3,006 | 3,024 | 3,042 | 3,06 | 3,078 | 18 ° | 3 | 6 | 9 |
72 ° | 3,078 | 3,096 | 3,115 | 3,133 | 3,152 | 3,172 | 3,191 | 3,211 | 3,230 | 3,251 | 3,271 | 17 ° | 3 | 6 | 10 |
73 ° | 3,271 | 3,291 | 3,312 | 3,333 | 3,354 | 3,376 | 3 | 7 | 10 | ||||||
3,398 | 3,42 | 3,442 | 3,465 | 3,487 | 16 ° | 4 | 7 | 11 | |||||||
74 ° | 3,487 | 3,511 | 3,534 | 3,558 | 3,582 | 3,606 | 4 | 8 | 12 | ||||||
3,630 | 3,655 | 3,681 | 3,706 | 3,732 | 15 ° | 4 | 8 | 13 | |||||||
75 ° | 3,732 | 3,758 | 3,785 | 3,812 | 3,839 | 3,867 | 4 | 9 | 13 | ||||||
3,895 | 3,923 | 3,952 | 3,981 | 4,011 | 14 ° | 5 | 10 | 14 | |||||||
tg | 60" | 54" | 48" | 42" | 36" | 30" | 24" | 18" | 12" | 6" | 0" | ctg | 1" | 2" | 3" |
How to use Bradis tables
Consider the Bradis table for sines and cosines. Everything related to sinuses is at the top and left. If we need cosines, we look at the right side at the bottom of the table.
To find the values of the sine of an angle, you need to find the intersection of the row containing the required number of degrees in the leftmost cell and the column containing the required number of minutes in the upper cell.
If the exact value of the angle is not in the Bradis table, we resort to the help of corrections. Corrections for one, two and three minutes are given in the right-most columns of the table. To find the value of the sine of an angle that is not in the table, we find the value closest to it. After that, add or subtract the correction corresponding to the difference between the angles.
If we are looking for the sine of an angle that is greater than 90 degrees, first we need to use the reduction formulas, and only then - the Bradis table.
Example. How to use the Bradis table
Suppose you need to find the sine of the angle 17 ° 44 ". According to the table, we find what the sine of 17 ° 42" is and add to its value a correction for two minutes:
17 ° 44 "- 17 ° 42" \ u003d 2 \ "(not about w o d i a i a i o r a y) sin 17 ° 44" \ u003d 0. 3040 + 0. 0006 = 0. 3046
The principle of working with cosines, tangents and cotangents is similar. However, it is important to remember about the sign of the amendments.
Important!
When calculating the values of sines, the correction has a positive sign, and when calculating the cosines, the correction must be taken with a negative sign.
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The main trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Based on this, the tangent of an angle in trigonometry is defined as a trigonometric function that expresses the ratio of the sine of this angle to the cosine of the same angle. If it is necessary to determine the tangent of an acute angle in a right-angled triangle, then it can be calculated geometrically, since the tangent in this case will be equal to the ratio of the opposite leg to the adjacent leg of the right-angled triangle. The term "tangent" itself is borrowed from Latin, its literal translation means “touching”. The tangent is indicated by Latin letters. The tangent of the x angle will be denoted "tg x", although Western mathematicians traditionally abbreviate the tangent to english word: the tangent of an angle x is referred to there as "tan x".
What is the tangent of 30 degrees
Based on the fact that the tangent of an angle is equal to the ratio of the sine of an angle to the cosine of the same angle, the tangent of an angle of 30 degrees can be obtained by dividing the value of the sine of an angle of 30 degrees by the value of the cosine of the same angle. The tangent will be 0.5774.
What is the tangent of 60 degrees
The tangent of an angle of 60 degrees is calculated in a similar way: dividing the sine of an angle of 60 degrees by the value of the cosine of the same angle gives the number 1.7321, which is the tangent of 60 degrees.
What is the tangent of 45 degrees
Since the sine value of a 45 degree angle is equal to the cosine value of the same angle, the 45 degree tangent value obtained by dividing the sine by cosine is one (the tangent is 1).
What is the tangent of 90 degrees
The tangent of an angle of 90 degrees cannot be calculated, since the cosine of an angle of 90 degrees is equal to zero, and one of the basic rules of division is the rule according to which "you cannot divide by zero", while the tangent in this case must be obtained by dividing the sine by cosine, that is, to zero. The tangent of 90 degrees is not specified.
What is the tangent of 120 degrees
Similarly, by calculating the tangent of 120 degrees, you can get the number -1.7321 (negative), which will be the tangent of 120 degrees.
What is the tangent of 0 degrees
Since the sine of an angle of 0 degrees is zero, and the cosine of the same angle is 1, the tangent is obtained by dividing zero by one, which gives 0. The tangent of 0 degrees is thus equal to 0.
What is the tangent of 135 degrees
The tangent of 135 degrees is -1 (minus one) by a similar calculation.