# Opposites divided by adjacent and the world of Tan

NOTE:

Only look at this if you want to know where cosine comes from.

Mathematicians built a world of right-angled triangles where the hypotenuse was always 1m long.

The mathematicians lifted a 1metre stick up in one degree intervals and measured how far the end lifted off the floor, and the adjacent length along the floor.

And for each degree they measured the length of both the opposite and the adjacent side of the triangle and plotted it on a graph. (The red triangle above shows the stick at 45 degrees. The adjacent side is 0.707 metres along the floor.)

They then divided the opposite distance by the adjacent distance and they called that the table/chart of tan (angle). It reads as follows:

 Angle Opposite distance (or sine) Adjacent distance (or cosine) (Opposite\ dist\ance\ (or sin))/(Adjacent\ dist\ance\ (or cos)) = tan 0^@ 0 \ metres 1 \ metre 0/1 =0 1^@ 0.01745 \ metres 0.99985 \ metres 0.01745/0.99985 =0.01746 2^@ 0.03490 \ metres 0.99939 \ metres 0.03490/0.99939 =0.03492 3^@ 0.05234 \ metres 0.99863 \ metres 0.05234/0.99863 =0.05241 4^@ 0.06976 \ metres 0.99756 \ metres 0.06976/0.99756 =0.06993 5^@ 0.08716 \ metres 0.99619 \ metres 0.08716/0.99619 =0.08749 6^@ 0.10453 \ metres 0.99452 \ metres 0.10453/0.99452 =0.10510 7^@ 0.12187 \ metres 0.99255 \ metres 0.12187/0.99255 =0.12278 8^@ 0.13917 \ metres 0.99027 \ metres 0.13917/0.99027 =0.14054 9^@ 0.15643 \ metres 0.98769 \ metres 0.15643/0.98769 =0.15838 10^@ 0.17365 \ metres 0.98481 \ metres 0.17365/0.98163 =0.17633 11^@ 0.19081 \ metres 0.98163 \ metres 0.19081/0.98163 =0.19438 12^@ 0.20791 \ metres 0.97815 \ metres 0.20791/0.97815 =0.21256 13^@ 0.22495 \ metres 0.97437 \ metres 0.22495/0.97437 =0.23087 14^@ 0.24192 \ metres 0.97030 \ metres 0.24192/0.97030 =0.24933 15^@ 0.25882 \ metres 0.96593 \ metres 0.25882/0.96593 =0.26795 16^@ 0.27564 \ metres 0.96126 \ metres 0.27564/0.96126 =0.28675 17^@ 0.29237 \ metres 0.95630 \ metres 0.29237/0.95630 =0.30573 18^@ 0.30902 \ metres 0.95106 \ metres 0.30902/0.95106 =0.32492 19^@ 0.32557 \ metres 0.94552 \ metres 0.32557/0.94552 =0.34433 20^@ 0.34202 \ metres 0.93969 \ metres 0.34202/0.93969 =0.36397 21^@ 0.35837 \ metres 0.93358 \ metres 0.35837/0.93358 =0.38386 22^@ 0.37461 \ metres 0.92718 \ metres 0.37461/0.92718 =0.40403 23^@ 0.39073 \ metres 0.92050 \ metres 0.39073/0.92050 =0.42447 24^@ 0.40674 \ metres 0.91355 \ metres 0.40674/0.91355 =0.44523 25^@ 0.42262 \ metres 0.90631 \ metres 0.42262/0.90631 =0.46631 26^@ 0.43837 \ metres 0.89879 \ metres 0.43837/0.89879 =0.48773 27^@ 0.45399 \ metres 0.89101 \ metres 0.45399/0.89101 =0.50952 28^@ 0.46947 \ metres 0.88295 \ metres 0.46947/0.88295 =0.53171 29^@ 0.48481 \ metres 0.87462 \ metres 0.48481/0.87462 =0.55431 30^@ 0.5 \ metres 0.86603 \ metres 0.5/0.86603 =0.57735 31^@ 0.51504 \ metres 0.85717 \ metres 0.51504/0.85717 =0.60086 32^@ 0.52992 \ metres 0.84805 \ metres 0.54464/0.84805 =0.62487 33^@ 0.54464 \ metres 0.83867 \ metres 0.54464/0.83867 =0.64941 34^@ 0.55919 \ metres 0.82904 \ metres 0.55919/0.82904 =0.67451 35^@ 0.57358 \ metres 0.81915 \ metres 0.57358/0.81915 =0.70021 36^@ 0.58779 \ metres 0.80901 \ metres 0.58779/0.80901 =0.72654 37^@ 0.60182 \ metres 0.79864 \ metres 0.60182/0.79864 =0.75355 38^@ 0.61566 \ metres 0.78801 \ metres 0.61566/0.78801 =0.78128 39^@ 0.62932 \ metres 0.77715 \ metres 0.62932/0.77715 =0.80978 40^@ 0.64279 \ metres 0.76604 \ metres 0.64279/0.76604 =0.83910 41^@ 0.65606 \ metres 0.75471 \ metres 0.65606/0.75471 =0.86928 42^@ 0.66913 \ metres 0.74314 \ metres 0.66913/0.74314 =0.90040 43^@ 0.68200 \ metres 0.73135 \ metres 0.68200/0.73135 =0.93251 44^@ 0.69466 \ metres 0.71934 \ metres 0.69466/0.71934 =0.96569 45^@ 0.70711 \ metres 0.70711 \ metres 0.70711/0.70711 =1 46^@ 0.71934 \ metres 0.69466 \ metres 0.71934/0.69466 =1.03553 47^@ 0.73135 \ metres 0.68200 \ metres 0.73135/0.68200 =1.07237 48^@ 0.74314 \ metres 0.66913 \ metres 0.74314/0.66913 =1.11061 49^@ 0.75471 \ metres 0.65606 \ metres 0.75471/0.65606 =1.15037 50^@ 0.76604 \ metres 0.64279 \ metres 0.76604/0.64279 =1.19175 51^@ 0.77715 \ metres 0.62932 \ metres 0.77715/0.62932 =1.23490 52^@ 0.78801 \ metres 0.61566 \ metres 0.78801/0.61566 =1.27994 53^@ 0.79864 \ metres 0.60182 \ metres 0.79864/0.60182 =1.32704 54^@ 0.80901 \ metres 0.58779 \ metres 0.80901/0.58779 =1.37638 55^@ 0.81915 \ metres 0.57358 \ metres 0.8195/0.57358 =1.42815 56^@ 0.82904 \ metres 0.55919 \ metres 0.82904/0.55919 =1.48256 57^@ 0.83867 \ metres 0.54464 \ metres 0.83867/0.54464 =1.53986 58^@ 0.84805 \ metres 0.52992 \ metres 0.84805/0.52992 =1.60033 59^@ 0.85717 \ metres 0.51504 \ metres 0.85717/0.51504 =1.66428 60^@ 0.86603 \ metres 0.5 \ metres 0.86603/0.5 =1.73205 61^@ 0.87462 \ metres 0.48481 \ metres 0.087462/0.48481 =1.80405 62^@ 0.88295 \ metres 0.46947 \ metres 0.88295/0.46947 =1.88073 63^@ 0.89101 \ metres 0.45399 \ metres 0.89101/0.45399 =1.96261 64^@ 0.89879 \ metres 0.43837 \ metres 0.89879/0.43837 =2.05030 65^@ 0.90631 \ metres 0.42262 \ metres 0.90631/0.42262 =2.14450 66^@ 0.91355 \ metres 0.40674 \ metres 0.91355/0.40674 =2.24604 67^@ 0.92050 \ metres 0.39073 \ metres 0.92050/0.39073 =2.35585 68^@ 0.92718 \ metres 0.37461 \ metres 0.92718/0.37461 =2.47509 69^@ 0.93358 \ metres 0.35837 \ metres 0.93358/0.35837 =2.60509 70^@ 0.93969 \ metres 0.34202 \ metres 0.93969/0.34202 =2.74747 71^@ 0.94552 \ metres 0.32557 \ metres 0.94552/0.32557 =2.90421 72^@ 0.95106 \ metres 0.30902 \ metres 0.95106/0.30902 =3.07768 73^@ 0.95630 \ metres 0.29237 \ metres 0.95630/0.29237 =3.27085 74^@ 0.96126 \ metres 0.27564 \ metres 0.96126/0.27564 =3.48741 75^@ 0.96593 \ metres 0.25882 \ metres 0.96593/0.25882 =3.73205 76^@ 0.97030 \ metres 0.24192 \ metres 0.97030/0.24192 =4.01078 77^@ 0.97437 \ metres 0.22495 \ metres 0.97437/0.22495 =4.33148 78^@ 0.97815 \ metres 0.20791 \ metres 0.97815/0.20791 =4.70463 79^@ 0.98163 \ metres 0.19081 \ metres 0.98163/0.19081 =5.14455 80^@ 0.98481 \ metres 0.17365 \ metres 0.98481/0.17365 =5.67128 81^@ 0.98769 \ metres 0.15643 \ metres 0.98769/0.15643 =6.31375 82^@ 0.99027 \ metres 0.13917 \ metres 0.99027/0.13917 =7.11537 83^@ 0.99255 \ metres 0.12187 \ metres 0.99255/0.12187 =8.14435 84^@ 0.99452 \ metres 0.10453 \ metres 0.99452/0.10453 =9.51436 85^@ 0.99619 \ metres 0.08716 \ metres 0.99619/0.08716 =11.43005 86^@ 0.99756 \ metres 0.06976 \ metres 0.99756/0.06976 =14.30067 87^@ 0.99863 \ metres 0.05234 \ metres 0.99863/0.05234 =19.08114 88^@ 0.99939 \ metres 0.03490 \ metres 0.99939/0.03490 =28.63625 89^@ 0.99985 \ metres 0.01745 \ metres 0.99985/0.01745 =57.28996 90^@ 1 \ metre 0 \ metres undefined undefined

Example 1

If the adjacent of a right angle triangle is 2 metres long and the opposite side is 1 metre long, what is the angle theta ?

Here we don’t know the hypotenuse, but we do know the length of the adjacent and opposite sides.

Simply divide the opposite by the adjacent length to calculate the unknown length.

i.e. 1/2=0.5

Look up on the table/chart of tan (angle) for 0.5 metres above.

 Angle (Opposite\ dist\ance\ (sin))/(Adjacent\ dist\ance\ (cos)) = tan ? 0.5

The nearest = 27°

But on a calculator if you put in 0.5 and press inverse tan (tan-1), this gives you 26.56°, which is more accurate.