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Let us say ABC is a right-angled triangle at B, such that;
∠ A = ∠ C = 45°
Thus, BC = AB = a (say)
Using Pythagoras theorem, we have;
AC
2
= AB
2
+ BC
2
= a
2
+ a
2
= 2a
2
AC = a√2
Now, from the trigonometric ratios, we have;
According to the congruency of the triangle, we can say;
Δ ABD ≅ Δ ACD
Hence,
BD = DC
∠ BAD = ∠ CAD (By CPCT)
Now, in triangle ABD, ∠ BAD = 30° and ∠ ABD = 60°
Using Pythagoras theorem,
AD
2
= AB
2
– BD
2
= (2a)
2
– (a)
2
= 3a
2
AD = a√3
So, the trigonometric ratios for a 30-degree angle will be;
sin 30° = BD/AB = a/2a = 1/2
cos 30° = AD/AB = a√3/2a = √3/2
tan 30° = BD/AD = a/a√3 = 1/√3
Also,
cosec 30° = 1/sin 30 = 2
sec 30° = 1/cos 30 = 2/√3
cot 30° = 1/tan 30 = √3
Similarly, we can derive the values of trigonometric ratios for 60°.
