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57. We have now virtually shown how also to trace . the variations in magnitude and sign of the cosecant, secant, and cotangent of an angle as the angle varies ; for these are easily deducible from those of the sine, cosine, and tangent respectively. For example, we may treat the secant thus::

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As A increases from 0 to 90°, cos A decreases and is positive; therefore sec A or

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increases and is

cos A

1

=

cos 90° 0

.. sec 90° = ∞.

As A increases from 90° to 180°, cos A increases and is negative; therefore sec A decreases and is negative. And so on throughout the four quadrants.

Again, the same variations for the cosecant, secant, and cotangent may be traced by means of a figure, exactly as those for the sine, cosine, and tangent have been traced.

58. General summary.-Let angle A pass continuously through all values from 0 to 360°; then

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59. It should be observed that 0 and o are points of transition from positive to negative and

from negative to positive for continuously varying quantities. Thus the preceding investigations show

that

sines and cosines change sign on passing through 0, tangents and cotangents on passing through 0 and ∞o, and secants and cosecants on passing through co.

Again, it is interesting to notice the limits of the trigonometrical ratios, and the manner in which each of them varies from one of its limits to the other.

The limits of the sine are 1 and -1; and it moves from the first to the second by decreasing, passing through 0 and changing sign, and increasing numerically. The limits of the cosecant are the same; but it may be said to pass from one to the other by a different route it increases, passes through co and changes sign, and decreases numerically. What has been said of sine and cosecant applies exactly to cosine and

secant.

The values of the tangent and cotangent have no limits. All values positive or negative are possible to

them.

60. (1.) What values of 0 less than 27 satisfy the equation

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:

2

cote (cot 0+1):

Therefore (i.) ... cot 0 = 0 = cot or cot

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(3.) Trace the variation in sign of sin A-cos A, as A increases from 0 to 360°.

When A is between 0 and 45°, cos ▲ is greater than sin A and is positive; therefore sin A-cos A is negative.

When A is between 45° and 135°, sin A is greater than cos A and positive; therefore sin A-cos A is positive.

When A is between 135° and 225°, cos A is greater than sin A and negative; therefore sin A-cos A is positive.

When A is between 225° and 315°, sin A is greater than cos A and negative; therefore sin A-cos A is negative.

When A is between 315° and 360°, cos A is greater than sin A and positive, therefore sin A-cos A is negative.

EXAMPLES.

1. Trace by means of a figure the variation in magnitude and sign of cot A, as A increases from 0 to 360°.

2. Prove geometrically that when A is less than 45°, sin A is less than cos A; and when A is between 45° and 135°, sin A is greater than cos A.

3. Show geometrically which is greater, tan A or cot A, when A is between 135° and 225°.

4. Show geometrically the absurdity of the equations

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5. A, is in the first quadrant, and A, in second quadrant, while both are very near to 90° in magnitude: compare tan ▲, and tan A2; also sec A, and sec A2.

6. For what values of A less than 360° is cot A very great in magnitude and negative in sign?

7. For what values of A less than 360° is cosec A a very large negative quantity ?

8. Deduce from a geometrical construction the range of the values of sec A, A being any angle.

9. What are the nearest approaches of the cosecant to zero in value ?

10. Show that the tangent may have any finite value, positive or negative. Is this also true of the cotangent and secant ?

:

Solve the following equations for values of less than 27 :11. sin2 0 = 1.

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14. tan20+√3 tan 0 = : 0.

15. 3 cosec e−(2+√√3) cosec 0 + 2 = 0.

Trace the changes in sign and value of the following expressions, as changes continuously from 0 to 2π:

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CHAPTER VIII.

GENERAL EXPRESSIONS FOR ANGLES HAVING
THE SAME SINE, COSINE, &c.

61. If not already familiar with them, the student should notice the following points in Algebra.

We commonly use n to represent any whole number whatever. Whenever such is its meaning, since the doubles of all numbers are necessarily even, all expressions like the following,

2(n-1), 2n, 2(n+1), &c., clearly represent even numbers.

On very little reflection it will appear that adding an odd number to or subtracting an odd number from an even number produces an odd number. Then expressions like the following

2n−1, 2n+1, 2n+3, &c.

must represent odd numbers.

Again, the value of the expression (-1)" is +1 when n is 0 or an even number, but -1 when n is an odd number; for

(−1)°=1, (−1)1=—1, (−1)2=1, (−1)3=—1,

and so on.

(-1)=1,

62. To find an expression for all angles which have a given value for sine.

Make AOP equal to P

the least positive angle

whose sine is the given

value; produce 40 to A', and make A'OP' equal to

AOP. Then it is easily AN'

shown that

sin AOP = sin A0P;

N Ꭺ

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