Now sin' and cos 0, being both squares, are both necessarily positive. Hence, since their sum is unity, neither of them can be greater than unity.

[For if one of them, say sin2 0, were greater than unity, the other, cos2 0, would have to be negative, which is impossible.]

Hence neither the sine nor the cosine can be numerically greater than unity.

Since sin cannot be greater than unity therefore

30. The foregoing results follow easily from the figure of Art. 23.

For, whatever be the value of the angle AOP, neither the side OM nor the side MP is ever greater than OP.

MP Since MP is never greater than OP the ratio is OP

never greater than unity, so that the sine of an angle is never greater than unity.

OM

Also since OM is never greater than OP, the ratio OP is never greater than unity, i.e. the cosine is never greater than unity.

31. We can express the trigonometrical ratios of an angle in terms of any one of them.

The simplest method of procedure is best shewn by examples.

Ex. 1. To express all the trigonometrical ratios in terms of the sine. Let AOP be any angle 0.

Let the length OP be unity and let the corresponding length of MP be s.

I

1-S2

M

A

By Euc. I. 47, OM = √OP2 – MP2 = √1 — s2.

The last five equations give what is required.

Ex. 2. To express all the trigonometrical relations in terms of the cotangent.

Taking the usual figure let the

length MP be unity, and let the corresponding value of OM be x.

By Euc. I. 47,

OP=NOM2 + MP2 = √1+x2.

√1+x2

The last five equations give what is required.

It will be noticed that, in each case, the denominator of the fraction which defines the trigonometrical ratio was

taken equal to unity. For example, the sine is

MP

OP

and

hence in Ex. 1 the denominator OP is taken equal to unity.

OM
MP'

The cotangent is and hence in Ex. 2 the side MP

is taken equal to unity.

Similarly suppose we had to express the other ratios in terms of the cosine, we should, since the cosine is equal OM

to put OP equal to unity and OM equal to c. The ОР

working would then be similar to that of Exs. 1 and 2.

In the following examples the sides have numerical values.

Ex. 3. If cos 0 equal

3

If cos 0 equal 5, find the values of the other ratios.

Along the initial line OA take OM equal to 3, and erect a perpendicular MP.

Let a line OP, of length 5, revolve round O until its other end meets this perpendicular in the point P. Then AOP is the angle 0.