by
computation rather than by measurement of segments. This is a
task which at the start of your study of mathematics you can only
execute for the angles of selected, simple triangles.Calculation of selected values of the trigonometric functions
We have not yet completely
solved the set task of computing from three given items the
remaining items of a triangle. We must still learn how to
determine values of the trigonometric functions for an arbitrary
angle by
computation rather than by measurement of segments. This is a
task which at the start of your study of mathematics you can only
execute for the angles of selected, simple triangles.
Isosceles, right-angled triangle
The sum of the angles at the
base of a right-angled, isosceles triangle is 90º, whence = 
= 45º. Let AC = a, then
| AB² = AC² + CB² | AB =  = a |
|
| sin 45º=a/a= = 1/ = /2 |
cos 45º = a/a=
= 1/ = /2 |
|
| tan 45º = a/a = 1 | cot 45º = a/a = 1 |
If you draw in any equilateral triangle the height CD on to AB, then, setting AC = a, AD = a/2 and by Pythagoras' Rule
You can now obtain the values of the trigonometric functions for 30º and 60º directly from the triangle ADC:
| sin 30º = (a/2)/a = 1/2 | sin 60º =(a /2)/a
= /2 |
|
| cos 30º = ( a/2)/a
= /2 |
cos 60º = (a/2)/a = 1/2 | |
| tan 30º=(a/2)/(a/2)=1/=/3 |
tan 60º = (a/2)/(a/2)
= |
|
| cot 60º = (a/2)/(a/2)
= |
cot 60º=(a/2)/(a/2)=1/=/3 |
In the triangles above, you can set a = 1.
sin 30º =  /2 |
cos 30º = /2 |
tan 30º = /3 |
cot 30º = |
|||
| sin 45º = /2 |
cos 45º = /2 |
tan 45º = 1 | cot 45º = 1 | |||
| sin 60º = /2 |
cos 60º = /2 |
tan 60º = |
cot 60º = /3 |
Further values
are listed in the next table. For the angles corresponding to the
column on the left hand side, the functions at the top apply, for
those on the right hand side, the functions at the bottom.
Task 1: Find the values of cos 40º, tan 70º, cot 30º.
cos40º=0.7660, tan70º=2.747, cot30º=1.732.
Task 2: Find from the given values: 1) cot = 1.192,
2) tan = 5.671,
3) cos =0.3420.
1) =
60º, 2) = 80º, 3) = 70º.