21. Fundamental relations between the functions of an From the figures of Art. 11, it is evident that for an

angle.

Also, from the definitions of the trigonometric functions, the following reciprocal relations are evident:

The eight formulas of this article are identities for they are true for any angle whatever. They are often spoken of as fundamental identities.

EXERCISES

In the following exercises determine the remaining functions from the given functions by means of the fundamental identities.

Note. The proper algebraic sign is determined by referring to the table of signs of Art. 13.

15. If tan ẞ = c, show that csc ẞ is real for all values of c.

22. To express one function in terms of each Suppose that we wish to express sin

functions.

each of the other functions.

of the other

in terms of

The algebraic sign of sin is determined from the quadrant in which ◊ is found.

23. To express all the functions of an angle in terms of one function of the angle, by means of a triangle. —

Example 1. Express all the functions of in terms of sin 0.

Solution. Construct angle 0 in the first quadrant, Fig. 28, and choose the point P in the terminal side with coördinates OM and

MP. Then by definition sin 0 =

MP

OP'

and if OP is taken equal

to 1, MP = sin 0, and OM = VOP2 - MP2 = √1 - sin2 0. The remaining functions may then be written as follows:

Example 2. Express all the functions in terms of cos 0.

Solution. Construct angle in the first quadrant, Fig. 29, and choose the point P in the terminal side with coördinates OM

equal to 1, OM = cos 0, and MP = √OP? – OM2

= √1 — cos2 0.

The remaining functions may then be written as follows:

In the following table, the student is asked to show that each function in the first column is equal to every expression found in the same row with the function.

The table has been prepared under the assumption that @ is an acute angle. Should be in any other quadrant, the proper sign for each function may then be determined.

24. Transformation of trigonometric expressions so as to contain but one function. In all transformations, avoid radicals if possible. If the expression is in a factored form, it is often desirable to reduce each factor separately and multiply the results.