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ON THE RELATIONS BETWEEN THE TRIGONOMETRICAL
RATIOS OF THE SAME ANGLE.
102. THE following relations are evident from the definitions : 1 1
1 cosec A
We have sin A
hypotenuse sin perpendicular
= tan . base
104. We may prove similarly cot 0
1 Or thus, cot 0 =
sin cos A sin 8°
105. Euclid I. 47 gives us that in any right-angled triangle the square on the hypotenuse the sum of the squares on the perpendicular and on the base,
or, (hypotenuse)' = (perpendicular) + (base).
(i) Divide each side of this identity by
(hypotenuse), and we get hypotenuse (perpendicular ,
base hypotenuse, hypotenuse hypotenuse that is,
1 = sino 0 + cos* 0.
(ii) Divide each side of the same identity by
(base)', and we get
(üi) Divide each side of the same identity by
(perpendicular)", and we get hypotenuse
) * = (perpendicular) base perpendicular perpendicular perpendicular)
(i) cos 0 + sin 0 = 1 (ii) 1 + tano 0 = sec 0
(iii) 1+ cot' 0 = cosec A are each a statement in Trigonometrical language of Euc. I. 47.
107. We give the above proof in a different form.
In OE take any point P, and draw PM perpendicular to OR. Then with respect to 0, MP is the perpendicular, OP is the hypotenuse, and OM is the base ;
OM .. sino 0
cosA OP2 :
We have to prove that sino 0 + cos 0 = 1,
MP2 OM that is, that
=1, OP2 OP2
MP2 + OM OP2 i.e. that
i.e. that MP + OMP=OP?. But this is true by Euclid I. 47.
Therefore cos 0 + sin’ 0 =1.
1 + tano 0 = sec 0, and that
1 + cota A = cosec? 0.
108. The following is a LIST OF FORMULÆ with which the student must make himself familiar :
109. In proving Trigonometrical identities it is often convenient to express the other Trigonometrical Ratios in terms of the sine and cosine.
Example. Prove that tan A +cot A=sec A. cosec A.
COS A cot A=
sin A' 1
1 sec A=
and cosec A=
COS A we have to prove that
sin A COS A'sin A'
1 COS A. sin A
cos A. sin A' and this is true, because sinA + cosA=1.
110. Sometimes it is more convenient to express all the other Trigonometrical Ratios in terms of the sine only, or in terms of the cosine only.
Example. Prove that sin4 8+2 sin? 8. cos? 0=1 - cos4 0.
Hence, putting 1 - cos? , and (i – cos? 6)2 for sin? A and sin4 0 respectively, we have to prove that
(1 - cos? )2 +2.(1 - cosa 0). cos? A=1 - cost 0,
1-2 cosa 0 + cos4 6+2 cos0 - 2 cos4 0=1 - cos4 0,
1-cos40=1 - cos4 ,
which is true.
This example may be proved directly, by reversing the steps of the above proof; thus
.: (1-2 cos? 0 + c054 ) + 2 cos' 0 - 2 cos4 0=1 - cost 0,
.: (1 – cosa 0)2 + 2 cos 0 (1 - cos20)=1 - cost 0,
.. (sin? 0)2 + 2 cos? 8. sinA=1 - cos 8. Q.E.D.
NOTE. (1 – cos 6) is called the versed sine of 0; it is abbreviated thus