upon

Hence when an angle is determined by its cosine, either of these two angles may be chosen unless some restriction is placed the choice. The most common restriction is that the angle shall be positive and less than 180°; that is, in the first or second quadrant. For every cosine there will be one and only one angle between the limits 0° and 180°.

The Tangent. Since two angles which differ by 180° have the same tangent, it follows that if a be an angle corresponding to any tangent, 180° + a will equally correspond to it. Hence there is always a choice between these two angles, unless some restriction is placed upon the angle.

The Cotangent. The cotangent being determined, like the tangent, by the intersection of the revolving side with a tangent line, every pair of angles corresponding to the same tangent will also correspond to the same cotangent. Hence a and 180° +α always have the same cotangent.

a or

The Secant. We readily see that the angles a and 360° a have the same secant. Hence if a be an angle corresponding to a given secant, 360° sponding to that same secant.

a will be another angle corre

The Cosecant. From the diagram (§ 27) it is easy to show that any two supplementary angles have the same cosecant.

EXERCISES.

What other angles have the same sines as the following? 1. 105°; 2. 185°; 3. 290°.

What other angles have the same cosines as the following? 5. 165°; 6. 320°.

4. 72°;

What other angles have the same tangents as the following? 8. 205°; 9. 355°.

7. 50°;

10. A pair of angles having the same sine differ by 24°. What angles are they?

11. A pair of angles having the same cosine differ by 110°. What angles are they?

NOTE. There are two pairs of angles which answer each of the two last questions.

Find two values of a from each of the following equations:

32. Extension to unlimited angles. Since when any integral number of circumferences is added to an angle its trigonometric functions remain unaltered, we must, to find the most general expression for the angle corresponding to a given function, add an arbitrary number of circumferences to the angles found in the last section. Then the most general expression for angles which have the same sine will be

in which n may take all integral values, positive and negative, including zero, and a must have such a value that 90-a shall correspond to the given sine.

This statement also means that all the angles formed by giving different values to n, while a remains constant, will have the same sine.

Example. Let us suppose the given sine to be that of 65°. Then a 25°, and the pairs of angles

find the four corresponding values of x within the first circumfer

find the four corresponding values of a.

First find the two values of 2c, and then, by SS 10-13, the two values of half

6. Show that if the value of tana be given there will be only one value of a to correspond to it within the first circumference. 7. If sin = sin 15°,

find what two values may have, and show that these values will have the same cosine.

8. Form the most general expression for all angles having the same cosine.

9. Form the most general expression for all angles having the same tangent.

Relations between the Six Trigonometric Functions.

33. When we know any one trigonometric function of an angle two definite values of the angle can then be determined, by constructions like those of $19. Knowing the angle, the values of the other functions can be found. Hence from any one function of an angle all the others can be found. We have now to investigate the algebraic relations by which this may be done, seeking to express each function in terms of the five others.

there will be two angles to correspond to it, of which one will be as much less than 90° as the other is greater. The cosines of these angles will be equal with opposite algebraic signs.

The similar triangles OPM and OXN give

OP: PM= 0X: XN;

OP: OM OX: ON.

Substituting for the lines their algebraic equivalents, the first proportion gives

which gives the tangent in terms of the sine and cosine.

The second proportion gives

(2)

(3)

We conclude from this:

The product of the cosine and secant is equal to unity.

In other words, the secant and cosine are reciprocals of each other.

By a similar course of reasoning upon the complementary triangle we find that the cosecant and sine are reciprocals of each other.

Comparing with (2) we have the relation: The product of the tangent and cotangent of any angle is unity.

In other words, the tangent and cotangent are reciprocals of each other.

We thus reach the general conclusion that the three complementary functions are each the reciprocal of one of the three other functions, namely:

=

1

secant

cosine

1. If sin y = 0,60, find cos y, tan y, cot y, sec y, and cosec y 2. Find the values of the same functions when cos y = 0.60. 3. Prove that the mean proportional between a cos x and a sec x is a.

4. Prove that the mean proportional between a tan ẞ and b cot B is Vab.

34. Expression of each function in terms of the others. By means of the relations (1) to (4) any one trigonometric function may be expressed in terms of any other by algebraic substitutions.

The following are the expressions which we thus obtain. All not already given should be deduced by the student as an exercise.