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§ 25. FUNCTIONS OF A VARIABLE ANGLE.

Let the angle x increase continuously from 0° to 360°; what changes will the values of its functions undergo?

It is easy, by reference to Figs. 21-24, to trace these changes throughout all the quadrants.

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1. The Sine. In the first quadrant, the sine MP increases from 0 to 1; in the second, it remains positive, and decreases from 1 to 0; in the third, it is negative, and increases in absolute value from 0 to 1; in the fourth, it is negative, and decreases in absolute value from 1 to 0.

2. The Cosine. In the first quadrant, the cosine OM decreases from 1 to 0; in the second, it becomes negative and increases in absolute value from 0 to 1; in the third, it is negative and decreases in absolute value from 1 to 0; in the fourth, it is positive and increases from 0 to 1.

3. The Tangent. In the first quadrant, the tangent AT increases from 0 to ∞o; in the second quadrant, as soon as the angle exceeds 90° by the smallest conceivable amount, the moving radius OP, prolonged in the direction opposite to that of OP, will cut AT at a point T situated very far below A; hence, the tangents of angles near 90° in the second quadrant have very large negative values. As the angle increases, the tangent AT continues negative but diminishes in absolute value. When x= 180°, then Tcoincides with A, and tan 180°

= 0. In the third quadrant, the tangent is positive and increases from 0 to ∞o; in the fourth, it is negative and decreases in absolute value from ∞ to 0.

4. The Cotangent. In the first quadrant, the cotangent BS decreases from ∞ to 0; in the second quadrant, it is negative and increases in absolute value from 0 to ∞o; in the third and fourth quadrants, it has the same sign, and undergoes the same changes as in the first and second quadrants respectively.

5. The Secant. In the first quadrant, the secant OT increases from 1 to ∞o; in the second quadrant, it becomes negative (being measured in the direction opposite to that of OP), and decreases in absolute value from ∞ to 1, so that sec 180°-1; in the third quadrant, it continues negative, and increases in absolute value from 1 to ∞; in the fourth quadrant, it is positive, and decreases from ∞ to 1.

6. The Cosecant. In the first quadrant, the cosecant OS decreases from oo to 1; in the second quadrant, it remains positive, and increases from 1 to ∞o; in the third quadrant, it becomes negative, and decreases in absolute value from ∞ to 1, so that csc 270° -1; in the fourth quadrant, it is negative, and increases in absolute value from 1 to co.

The limiting values of the functions are as follows:

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Sines and cosines extend from +1 to

-1; tangents and cotangents from +∞ to -∞; secants and cosecants from +∞ to +1, and from -1 to

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In the table given above the double sign ± is placed before 0 and ∞. From the preceding investigation it appears that the functions always change sign in passing through 0 and ∞; and the sign + or - prefixed to 0 or simply shows the direction from which the value is reached. Take, for example, tan 90°: The nearer an acute angle is to 90°, the greater the positive value of its tangent; and the nearer an obtuse angle is to 90°, the greater the negative value of its tangent. When the angle is 900, OP (Fig. 21) is parallel to AT, and cannot meet it. But tan 90° may be regarded as extending either in the positive or in the negative direction; and according to the view taken, it will be + ∞ or -8.

§ 26. FUNCTIONS OF ANGLES LARGER THAN 360°. It is obvious that the functions of 360° +x are the same both in sign and in absolute value as those of x; for the moving radius has the same position in both cases. In general, if n denote any positive whole number,

=

The functions of (n × 360° +x) are the same as those of x. For example: the functions of 2200° the functions of (6 × 360° +40°)= the functions of 40°.

§ 27. EXTENSION OF FORMULAS [1]-[3] TO ALL ANGLES. The Formulas established for acute angles in § 5 hold true for all angles. Thus, Formula [1],

sinx+cosx=1,

is universally true; for, whether MP and OM (Figs. 21-24) are positive or negative, MP and OM' are always positive, and in each quadrant MP2+OM2=OP2=1.

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are universally true; for the algebraic signs of the functions, as given in the table at the end of § 24, agree with those in Formulas [2] and [3]; and with regard to the absolute values, we have in each quadrant from the similar triangles OMP, OAT, OBS, (Figs. 21-24) the proportions

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which, by substituting 1 for the radius, and the right names for the other lines, are easily reduced to the above formulas.

Formulas [1]-[3] enable us, from a given value of one function, to find the absolute values of the other five functions, and also the sign of the reciprocal function. But in order to determine the proper signs to be placed before the other four functions, we must know the quadrant to which the angle in question belongs; or what amounts to the same thing, the sign of any one of these four functions; for, by reference to the Table of Signs (§ 24) it will be seen that the signs of any two functions that are not reciprocals determine the quadrant to which the angle belongs.

EXAMPLE. Given sin x=+, and tan x negative; find the values of the other functions.

Since sinx is positive, x must be an angle in Quadrant I. or in Quadrant II.; but, since also tanx is negative, Quadrant I. is inadmissable.

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Since the angle is in Quadrant II. the minus sign must be taken, and we have

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1. Construct the functions of an angle in Quadrant II. What are their signs?

2. Construct the functions of an angle in Quadrant III. What are their signs?

3. Construct the functions of an angle in Quadrant IV. What are their signs?

4. What are the signs of the functions of the following angles: 340°, 239°, 145°, 400°, 700°, 1200°, 3800°?

5. How many angles less than 360° have the value of the sine equal to +, and in what quadrants do they lie?

6. How many values less than 720° can the angle x have if cosx+, and in what quadrants do they lie?

7. If we take into account only angles less than 180°, how many values can x have if sin x=? if cosx=? if cosx= -? if tanx=? if cotx=-7?

8. Within what limits must the angle x lie if cos x=? if cotx=4? if sec x 80? if csc x=- -3? (x to be less than

360°.)

9. In what quadrant does an angle lie if sine and cosine are both negative? if cosine and tangent are both negative? if the cotangent is positive and the sine negative?

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