Trigonometry


   
 
Trigonometrical Ratios of Standard Angles
T-ratios of standard angles
0o, 30o, 45o, 60o and 90o are called standard angles.
 
These angles are called standard angles because it is possible to obtain simple mathematical ratios for these angles.
 
The students are advised to learn the T-ratios of these angles, they can also obtain them as shown in the derivation.
 
T-ratios of 45o
 
Let
 
 
\ AB = BC = a
 
Using Pythagoras Theorem
 
AC2 = a2 + a2 = 2a2
 
 
 
 
 
 
 
 
 
T-ratios of 30o and 60o
 
Let and
 
 
In 30o - 60o - 90o triangle, it can be proved that the hypotenuse is double the side opposite to 30o (see proof in Geometry Section),
 
AC = 2AB
 
Let AB = a
 
AC = 2a
 
Using Pythagoras Theorem
 
BC2 = (2a)2 - a2
 
= 4a2 - a2
 
= 3a2
 
 
 
 
 
 
 
 
T-ratios of 0o and 90o
 
Draw a circle with radius r and XOX' and YOY' as axes.
 
 
Let
 
 
Let
 
 
 
 
We observe A"B" > A'B' > AB, denominator remains 'r' (radius).
 
As the angle increases sine ratio increases.
 
 
We observe OB'' < OB' < OB, denominator remains r.
 
As the angle increases cosine ratio decreases,
 
When a = 0,
 
 
 
 
 
When
 
 
 
 
(infinity)
 
As the angle increases tangent ratio increases.
 
 
From the above results we conclude that for an acute angle q, the following results hold good.
 
(i)
 
(ii)
 
(iii)
 
Aid to Memory
 
 
 
Aid to memorise the table for standard angles:
 
(a) Write 0, 1, 2, 3, 4 over each column as shown above.
 
(b) Divide each number by 4 and take the square root. The values obtained are sine ratios.
 
(c) Write these ratios in reverse order and you obtain the cosine ratios
 
(sin A = cos B if A + B = 90o)
 
(d) Divide each sine ratio by cosine ratio and you obtain the values of corresponding tangent ratios
 
 
 
Without using tables, find the value of
 
 
 
 
 
 
If A = 30o, verify sin 2A = 2 sin A cos A.
 
 
L.H.S sin 2A = sin 2 x 30
 
= sin 60
 
 
R.H.S 2 sin A cos A = 2 sin 30 cos 30
 
 
 
sin 2A = 2 sin A cos A
 
Hence L.H.S. = R.H.S.
 
 
     
   
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