With the reductions from the previous topic, it is enough to know the functions in the 1st quadrant. A final symmetry — in angle bisector of the 1st quadrant — reduces the work even further: it is enough to know the arcs from $0$ to $\frac{\pi}{4}$, that is, from $0^\circ$ to $45^\circ$.

The story

The first trigonometric tables were strings: Hipparchus and Ptolemy tabulated the string of each bow, half a degree by half. The Indians began to tabulate the half chord (the sine), and the Arab astronomers added tangents and cotangents, linked to the length of the shadows on sundials. In the 15th century, Regiomontanus calculated sines with 7 digits, and in 1596 the posthumous work of Rheticus, completed by Valentin Otho, brought the six functions of 10 in 10 seconds of arc, with up to 10 numerals — a decades-long work that calculators made unnecessary, but which underpinned navigation and astronomy for centuries. The symmetry of this topic was the first trick of those who assembled a board: calculating the sine up to $45^\circ$, the rest came for free.

  1. c. 140 a.C.
    Hiparchus makes the first known string board.
  2. c. 150
    Ptolemy: half-degree chords, in base 60.
  3. 499
    Aryabhata: 24 half-strings (sines), from $3^\circ 45'$ to $3^\circ 45'$.
  4. century. X
    Abu al-Wafa: sines from $15'$ to $15'$, with eight places.
  5. c. 1464
    Regiomontanus: sines with 7 digits.
  6. 1596
    Rheticus and Otho: the six functions from $10''$ to $10''$, with up to 10 digits.
  7. 1614
    Napier publishes tables of logarithms of sines.
  8. 1972
    Pocket scientific calculator retires the boards.

Sources: Regiomontanus (MacTutor) · Rheticus (MacTutor).

In everyday life

  • Printed tables. The old navigation tables only had the angles from $0^\circ$ to $45^\circ$: they were read from top to bottom for these angles and from bottom to top, with the names changed (sine ↔ cosine), for those from $45^\circ$ to $90^\circ$.
  • Ladder on the wall. If the ladder makes an angle $\alpha$ with the floor, it makes $90^\circ - \alpha$ with the wall; the height reached is $L\operatorname{sen}\alpha = L\cos(90^\circ - \alpha)$.
  • Height of the Sun. Astronomers use both the height of the Sun above the horizon and the distance to the zenith, which is its complement; the sine of one is the cosine of the other.
A staircase makes an angle with the wall that is complementary to that of the floor: the sine of one is the cosine of the other.

Theory

Reduction from [π/4, π/2] to [0, π/4]

Given $x$ with $\frac{\pi}{4} < x < \frac{\pi}{2}$, let $P$ be its image and $P'$ be the point of the symmetric cycle of $P$ in relation to angle bisector of the 1st quadrant. Counterclockwise, $\overset{\frown}{AP} + \overset{\frown}{PB} = \frac{\pi}{2}$; as $\overset{\frown}{PB} = \overset{\frown}{AP'}$, comes $\overset{\frown}{AP} + \overset{\frown}{AP'} = \frac{\pi}{2}$, then $\overset{\frown}{AP'} = \frac{\pi}{2} - x$.

Drag $P$: the point $P'$ is symmetric with respect to angle bisector of the 1st quadrant. The sine of one is the cosine of the other.

Due to the congruence of the triangles $OPP_2$ and $OP'P_1'$ (symmetrical with respect to angle bisector), we have $\overline{OP_2} = \overline{OP_1'}$ and $\overline{P_2P} = \overline{P_1'P'}$, that is:

PropriedadeComplementary arcs

$$\operatorname{sen} x = \cos\left(\frac{\pi}{2} - x\right) \qquad \cos x = \operatorname{sen}\left(\frac{\pi}{2} - x\right)$$

and, consequently,

$$\operatorname{tg} x = \frac{\cos\left(\frac{\pi}{2} - x\right)}{\operatorname{sen}\left(\frac{\pi}{2} - x\right)} = \operatorname{cotg}\left(\frac{\pi}{2} - x\right) \qquad \operatorname{cotg} x = \operatorname{tg}\left(\frac{\pi}{2} - x\right)$$$$\sec x = \operatorname{cossec}\left(\frac{\pi}{2} - x\right) \qquad \operatorname{cossec} x = \sec\left(\frac{\pi}{2} - x\right).$$

That’s why the cosine, cotangent and cosecant are called cofunctions: each one is the function of the complement.

ExemploExamples

$$\operatorname{sen} 71^\circ = \cos(90^\circ - 71^\circ) = \cos 19^\circ \qquad \cos 60^\circ = \operatorname{sen}(90^\circ - 60^\circ) = \operatorname{sen} 30^\circ \qquad \operatorname{tg} 50^\circ = \operatorname{cotg} 40^\circ$$$$\operatorname{sen}\frac{\pi}{3} = \cos\frac{\pi}{6} \qquad \cos\frac{5\pi}{12} = \operatorname{sen}\frac{\pi}{12} \qquad \operatorname{tg}\frac{3\pi}{8} = \operatorname{cotg}\frac{\pi}{8}$$

Table of values ​​of trigonometric functions

Thanks to cofunctions, a single table from $0^\circ$ to $45^\circ$ serves for $0^\circ$ to $90^\circ$: for the angles from $0^\circ$ to $45^\circ$, read the columns from top to bottom using the top header; for those $45^\circ$ to $90^\circ$, read from bottom to top using the bottom header. For example, $\operatorname{sen} 60^\circ$ is in the row of $30^\circ$, in the column whose bottom label is “sine”: $0{,}8660$.

Open the board (0° to 90°)
degrees radians sine tangent cotangent cosine radians degrees
0° 0,0000 0,0000 0,0000 — 1,0000 1,5708 90°
1° 0,0175 0,0175 0,0175 57,290 0,9998 1,5533 89°
2° 0,0349 0,0349 0,0349 28,636 0,9994 1,5359 88°
3° 0,0524 0,0523 0,0524 19,081 0,9986 1,5184 87°
4° 0,0698 0,0698 0,0699 14,301 0,9976 1,5010 86°
5° 0,0873 0,0872 0,0875 11,430 0,9962 1,4835 85°
6° 0,1047 0,1045 0,1051 9,5144 0,9945 1,4661 84°
7° 0,1222 0,1219 0,1228 8,1443 0,9925 1,4486 83°
8° 0,1396 0,1392 0,1405 7,1154 0,9903 1,4312 82°
9° 0,1571 0,1564 0,1584 6,3138 0,9877 1,4137 81°
10° 0,1745 0,1736 0,1763 5,6713 0,9848 1,3963 80°
11° 0,1920 0,1908 0,1944 5,1446 0,9816 1,3788 79°
12° 0,2094 0,2079 0,2126 4,7046 0,9781 1,3614 78°
13° 0,2269 0,2250 0,2309 4,3315 0,9744 1,3439 77°
14° 0,2443 0,2419 0,2493 4,0108 0,9703 1,3265 76°
15° 0,2618 0,2588 0,2679 3,7321 0,9659 1,3090 75°
16° 0,2793 0,2756 0,2867 3,4874 0,9613 1,2915 74°
17° 0,2967 0,2924 0,3057 3,2709 0,9563 1,2741 73°
18° 0,3142 0,3090 0,3249 3,0777 0,9511 1,2566 72°
19° 0,3316 0,3256 0,3443 2,9042 0,9455 1,2392 71°
20° 0,3491 0,3420 0,3640 2,7475 0,9397 1,2217 70°
21° 0,3665 0,3584 0,3839 2,6051 0,9336 1,2043 69°
22° 0,3840 0,3746 0,4040 2,4751 0,9272 1,1868 68°
23° 0,4014 0,3907 0,4245 2,3559 0,9205 1,1694 67°
24° 0,4189 0,4067 0,4452 2,2460 0,9135 1,1519 66°
25° 0,4363 0,4226 0,4663 2,1445 0,9063 1,1345 65°
26° 0,4538 0,4384 0,4877 2,0503 0,8988 1,1170 64°
27° 0,4712 0,4540 0,5095 1,9626 0,8910 1,0996 63°
28° 0,4887 0,4695 0,5317 1,8807 0,8829 1,0821 62°
29° 0,5061 0,4848 0,5543 1,8040 0,8746 1,0647 61°
30° 0,5236 0,5000 0,5774 1,7321 0,8660 1,0472 60°
31° 0,5411 0,5150 0,6009 1,6643 0,8572 1,0297 59°
32° 0,5585 0,5299 0,6249 1,6003 0,8480 1,0123 58°
33° 0,5760 0,5446 0,6494 1,5399 0,8387 0,9948 57°
34° 0,5934 0,5592 0,6745 1,4826 0,8290 0,9774 56°
35° 0,6109 0,5736 0,7002 1,4281 0,8192 0,9599 55°
36° 0,6283 0,5878 0,7265 1,3764 0,8090 0,9425 54°
37° 0,6458 0,6018 0,7536 1,3270 0,7986 0,9250 53°
38° 0,6632 0,6157 0,7813 1,2799 0,7880 0,9076 52°
39° 0,6807 0,6293 0,8098 1,2349 0,7771 0,8901 51°
40° 0,6981 0,6428 0,8391 1,1918 0,7660 0,8727 50°
41° 0,7156 0,6561 0,8693 1,1504 0,7547 0,8552 49°
42° 0,7330 0,6691 0,9004 1,1106 0,7431 0,8378 48°
43° 0,7505 0,6820 0,9325 1,0724 0,7314 0,8203 47°
44° 0,7679 0,6947 0,9657 1,0355 0,7193 0,8029 46°
45° 0,7854 0,7071 1,0000 1,0000 0,7071 0,7854 45°
cosine cotangent tangent sine radians degrees

AtençãoUsing the reduction board

To calculate $\cos 2861^\circ$: $2861^\circ - 7 \cdot 360^\circ = 341^\circ$ (4th quadrant), $\cos 341^\circ = \cos(360^\circ - 341^\circ) = \cos 19^\circ \approx 0{,}9455$.

The exercise in this section is number 118 from the chapter list.