Math Tables

Trigonometric Table

A trigonometric table is the original lookup table. Before calculators, every engineer carried a book of sines and interpolated between printed rows, and the tables were accurate to seven figures because that was what surveying needed. The values here are computed from Math.sin and Math.cos at build time, so they carry the same accuracy without the transcription risk.

Last updated 25 October 2026

Three things about the shape of the functions are worth internalising. Sine and cosine are the same curve shifted by 90 degrees, which is why sin 30 equals cos 60. Both stay between 0 and 1 in the first quadrant. And the tangent, which is sine divided by cosine, runs from 0 to infinity as the angle approaches 90, because the adjacent side shrinks to nothing while the opposite side grows.

That behaviour at 90 degrees is why tangent tables are the tricky ones. In floating point, tan(90°) does not evaluate to infinity; it evaluates to about 1.6 × 10^16 because π/2 cannot be represented exactly in binary. A table that prints that number instead of undefined is telling you more about floating point than about trigonometry.

The last table converts between the three ways slope is expressed. Builders talk in degrees, road engineers talk in percentage grade, and drainage codes talk in 1-in-n. They are all the tangent: a 45 degree slope is 100%, which is 1 in 1, because rise equals run. A 5% grade is 2.86 degrees, not 5 degrees, and that confusion has caused more than one drainage problem.

Sine, cosine and tangent every five degrees

Sine, cosine and tangent every five degrees
Degrees	Sine	Cosine	Tangent
0	0.000000	1.000000	0.000000
5	0.087156	0.996195	0.087489
10	0.173648	0.984808	0.176327
15	0.258819	0.965926	0.267949
20	0.342020	0.939693	0.363970
25	0.422618	0.906308	0.466308
30	0.500000	0.866025	0.577350
35	0.573576	0.819152	0.700208
40	0.642788	0.766044	0.839100
45	0.707107	0.707107	1.000000
50	0.766044	0.642788	1.191754
55	0.819152	0.573576	1.428148
60	0.866025	0.500000	1.732051
65	0.906308	0.422618	2.144507
70	0.939693	0.342020	2.747477
75	0.965926	0.258819	3.732051
80	0.984808	0.173648	5.671282
85	0.996195	0.087156	11.430052
90	1.000000	0.000000	∞

Sine rises from 0 to 1 and cosine falls from 1 to 0, meeting at 45 degrees where both are 0.707107. The tangent goes to infinity at 90 because the adjacent side collapses to zero.

Every degree from 0 to 45

Every degree from 0 to 45
Degrees	Sine	Cosine	Tangent
0	0.000000	1.000000	0.000000
1	0.017452	0.999848	0.017455
2	0.034899	0.999391	0.034921
3	0.052336	0.998630	0.052408
4	0.069756	0.997564	0.069927
5	0.087156	0.996195	0.087489
6	0.104528	0.994522	0.105104
7	0.121869	0.992546	0.122785
8	0.139173	0.990268	0.140541
9	0.156434	0.987688	0.158384
10	0.173648	0.984808	0.176327
11	0.190809	0.981627	0.194380
12	0.207912	0.978148	0.212557
13	0.224951	0.974370	0.230868
14	0.241922	0.970296	0.249328
15	0.258819	0.965926	0.267949
16	0.275637	0.961262	0.286745
17	0.292372	0.956305	0.305731
18	0.309017	0.951057	0.324920
19	0.325568	0.945519	0.344328
20	0.342020	0.939693	0.363970
21	0.358368	0.933580	0.383864
22	0.374607	0.927184	0.404026
23	0.390731	0.920505	0.424475
24	0.406737	0.913545	0.445229
25	0.422618	0.906308	0.466308
26	0.438371	0.898794	0.487733
27	0.453990	0.891007	0.509525
28	0.469472	0.882948	0.531709
29	0.484810	0.874620	0.554309
30	0.500000	0.866025	0.577350
31	0.515038	0.857167	0.600861
32	0.529919	0.848048	0.624869
33	0.544639	0.838671	0.649408
34	0.559193	0.829038	0.674509
35	0.573576	0.819152	0.700208
36	0.587785	0.809017	0.726543
37	0.601815	0.798636	0.753554
38	0.615661	0.788011	0.781286
39	0.629320	0.777146	0.809784
40	0.642788	0.766044	0.839100
41	0.656059	0.754710	0.869287
42	0.669131	0.743145	0.900404
43	0.681998	0.731354	0.932515
44	0.694658	0.719340	0.965689
45	0.707107	0.707107	1.000000

The first half of the quadrant. Small angles are where the approximation sin x ≈ x is accurate: at 5 degrees the sine is 0.0872 against a radian value of 0.0873, an error of 0.13%.

Every degree from 46 to 90

Every degree from 46 to 90
Degrees	Sine	Cosine	Tangent
46	0.719340	0.694658	1.035530
47	0.731354	0.681998	1.072369
48	0.743145	0.669131	1.110613
49	0.754710	0.656059	1.150368
50	0.766044	0.642788	1.191754
51	0.777146	0.629320	1.234897
52	0.788011	0.615661	1.279942
53	0.798636	0.601815	1.327045
54	0.809017	0.587785	1.376382
55	0.819152	0.573576	1.428148
56	0.829038	0.559193	1.482561
57	0.838671	0.544639	1.539865
58	0.848048	0.529919	1.600335
59	0.857167	0.515038	1.664279
60	0.866025	0.500000	1.732051
61	0.874620	0.484810	1.804048
62	0.882948	0.469472	1.880726
63	0.891007	0.453990	1.962611
64	0.898794	0.438371	2.050304
65	0.906308	0.422618	2.144507
66	0.913545	0.406737	2.246037
67	0.920505	0.390731	2.355852
68	0.927184	0.374607	2.475087
69	0.933580	0.358368	2.605089
70	0.939693	0.342020	2.747477
71	0.945519	0.325568	2.904211
72	0.951057	0.309017	3.077684
73	0.956305	0.292372	3.270853
74	0.961262	0.275637	3.487414
75	0.965926	0.258819	3.732051
76	0.970296	0.241922	4.010781
77	0.974370	0.224951	4.331476
78	0.978148	0.207912	4.704630
79	0.981627	0.190809	5.144554
80	0.984808	0.173648	5.671282
81	0.987688	0.156434	6.313752
82	0.990268	0.139173	7.115370
83	0.992546	0.121869	8.144346
84	0.994522	0.104528	9.514364
85	0.996195	0.087156	11.430052
86	0.997564	0.069756	14.300666
87	0.998630	0.052336	19.081137
88	0.999391	0.034899	28.636253
89	0.999848	0.017452	57.289962
90	1.000000	0.000000	∞

Above 45 degrees the tangent grows faster than the angle. At 89 degrees it is already 57.29, which is why a one degree error near vertical throws a plumb line far off.

Slope as an angle and as a percentage

Slope as an angle and as a percentage
Degrees	Slope (%)	Slope (1 in n)	Typical use
1	1.75	1 in 57.29	Drainage, road crown
2	3.49	1 in 28.64	Drainage, road crown
3	5.24	1 in 19.08	Ramps, driveways
4	6.99	1 in 14.3	Ramps, driveways
5	8.75	1 in 11.43	Ramps, driveways
6	10.51	1 in 9.51	Ramps, driveways
8	14.05	1 in 7.12	Ramps, driveways
10	17.63	1 in 5.67	Steep road
12	21.26	1 in 4.7	Steep road
15	26.79	1 in 3.73	Steep road
20	36.4	1 in 2.75	Roof pitch
25	46.63	1 in 2.14	Roof pitch
30	57.74	1 in 1.73	Roof pitch
35	70.02	1 in 1.43	Stairs, ladders
40	83.91	1 in 1.19	Stairs, ladders
45	100	1 in 1	Stairs, ladders

A 45 degree slope is 100%, because the rise equals the run. Road gradients above 10% are rare and above 20% are exceptional — the steepest paved street in the world is about 35%.

How to Use These Tables

Solving a right triangle: multiply the hypotenuse by the sine of the opposite angle to get the opposite side. A 10 m ladder at 70 degrees reaches 9.397 m up the wall.
Converting a road grade: take the arctangent of the percentage divided by 100. A 6% grade is 3.43 degrees, which is why it does not feel like much until you are climbing it.
Checking a roof: a 4-in-12 pitch is 18.43 degrees, because the tangent is 4/12 = 0.3333.
Drainage: a 1 in 60 fall is 1.67%, or 0.95 degrees. Codes often specify 1 in n rather than a percentage because it is easier to measure with a level and a rule.
Small angle approximation: below about 10 degrees, sin x ≈ x in radians within 0.5%. That is why the arc length and the chord are treated as equal in surveying.

Common Questions

Why is tan 90 undefined?

Because tangent is opposite over adjacent, and at 90 degrees the adjacent side has length zero. Division by zero has no value, so the tangent grows without bound as the angle approaches 90.

Is sin 30 exactly 0.5?

Yes, in exact geometry. The table prints 0.500000 because that is the rounded value; the exact value is 1/2, which is one of the few trigonometric values that is a rational number.

How do I convert a slope percentage to degrees?

Take the arctangent of the percentage divided by 100, then convert radians to degrees. 12% is atan(0.12) = 6.84 degrees.

What is the difference between sine and cosine?

A 90 degree phase shift. Cosine of an angle equals the sine of the angle plus 90 degrees, which is why cos 0 is 1 while sin 0 is 0.

Why do radians matter for calculus?

Because the derivative of sine is cosine only when the input is in radians. In degrees the derivative picks up a factor of π/180, which makes every formula messier.

What is a 1 in 40 fall in degrees?

1.43 degrees, or 2.5%. It is a common minimum for drainage pipes, and it is expressed as a ratio because it is easier to check on site with a level.

More Math Tables

Sources

NIST Digital Library of Mathematical Functions — trigonometric functions — National Institute of Standards and Technology
ISO 80000-2 — quantities and units, mathematics — International Organization for Standardization
AASHTO Green Book — geometric design of highways, grades and slopes — American Association of State Highway and Transportation Officials

Filed with Math calculations.