Make your own free website on Tripod.com

Contents Mathematics Fermat Physics Wordlist Notes Back Quick Facts Reference LINKS


The surd form of trigonometrical ratios:


Equalateral triangle of side 2 units and angles 60 degrees

The equalateral triangle above has sides each 2 units long and all angles at 60o. It has been halved into 2 right - angled triangles of base 1 unit long. Using Pythagorus:

(ab)2 = (ac)2 + (bc)2
=> (2)2 = (ac)2 + (1)2
=> (ac)2 = 4 - 1 = 3
and (ac) = root 3.


Using the definitions of trigonometrical ratios, summarised in SOHCAHTOA:

sin 30o = bc / ab = 1 / 2
sin 60o = ac / ab = root 3 / 2
cos 30o = ac / ab = root 3 / 2
cos 60o = bc / ab = 1 /2
tan 30o = bc / ac = 1 / root 3
tan 60 = ac / bc = root 3 / 1 = root 3


Isosceles of equal side 1 unit

The isosceles above has equal sides of 1 unit each and the 2 complementary angles each of 45o. Using Pythagoras' Theorem to find side (ac):

(ac)2 = (ab)2 + (bc)2
(ac)2 = (1)2 + (1)2 = 2
ac = root 2

Trigonometrical ratio: Surd form: Approximation
sin 30o 1 / 2 0.5
sin 45o 1 / root 2 0.7071
sin 60o root 3 / 2 0.866
cos 30o root 3 / 2 0.866
cos 45 1 / root 2 0.7071
cos 60o 1 / 2 0.5
tan 30o 1 / root 3 0.5774
tan 45o 1  
tan 60o root 3 1.7321

For any pair of complementary angles these 2 rules apply:

sin xo = cos (90 - x)
and cos xo = sin (90 - x)

sin 30o = cos 60o
sin 60o = cos 30o
sin 45o = cos 45o
sin 20o = cos 70o
cos 41o = sin 49o
cos 72o = sin 18o
sin 36o = cos 54o


2 similar triangles

When the angle is 30o on a right - angled triangle, then the side opposite the 30o angle is half the hypotenuse. This is also true for similar triangles. Sides H, A and O can be any value and O = 1 / 2 H. Therefore, sin 30o = 0.5.


Contents Mathematics Fermat Physics Wordlist Notes Back Quick Facts Reference LINKS