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Let OP, OP, OP,, OP, represent the position of the revolving line at any period of revolution in the several quadrants respectively,

And let P, N, P,Ň, P.N., PAN, be the respective perpendiculars from the end of the revolving line upon the initial line.

Then P,N,, P,N,, P,N,, PLN, are respectively the perpendiculars corresponding to the angles generated.

Also, ON, ON, ON, ON, are respectively the bases of the right-angled triangles with respect to the angles in question. We have then in the second quadrantSin AOP, = 22, cos AOP, = tan AOP ,, =

ON2 It is therefore evident that the relations between the trigonometrical ratios, which were proved to exist in Art. 7, also hold for angles in the second quadrant—that is, angles between 90° and 180°.

And in the same way we may show that they hold for angles in the third, fourth, or any quadrant.

And again, if we suppose the line to revolve in a negative direction, and take the position OP', we shall have P N the perpendicular corresponding to the negative angle AOP', and ON' the base.

Opis cos AOF, = OP

P'N'

tan AOP' = PN, &c. And the relations proved in Art. 7 may be also similarly proved to exist here.

Hence the relations proved in Art. 7 hold for any angles whatever. Changes of Magnitude and Sign of the Trigonometrical

Ratios. 11. Let OP,, OP, OP3, OP, be positions of the revolving line in the several quadrants respectively; P N, P,N,

Hence, sin AOP, _ P'N'

Opis cos AOP! - ON

P.N3, P.NĄ, the respective perpendiculars; and ON, ON2, ON, ON the bases of the corresponding right-angled triangles.

Then-
(1.) In the first quadrant-

Sin AOP, = B,D, cos AOP, = 9

ON

OP,

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At the commencement of the motion of the revolving line, the angle AOP, = 0°;

Also, the perpendicular P,N, = 0,
And the base ON = OP :
Hence, we have

* Sin 0° = o p = 0, cos 0o = OPI = 1,

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As the revolving line moves from OA towards OB, P,N, increases and ON, diminishes; and when it arrives at OB, we have P N = OP ,and ON, = 0. But the angle generated is now a right angle. Hence we haveSin 90° = op = 1, cos 90° = p = 1,

tan 90o = 081 = Co.
Hence, as the angle increases from 0° to 90°--
The sine changes in magnitude from 0 to 1 and is +.
The cosine changes in magnitude from 1 to 0 and is +.
The tangent changes in magnitude from 0 to co and is +.

(2.) In the second quadrant-
Here the perpendicular P,N, is +,

and the base ON is - .

* The student ought properly to look upon the values 0, 1, 0 here obtained as the limiting values of the sine, cosine, and tangent respectively, when the angle is indefinitely diminished.

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= 0, &c.

Hence the sine during the second quadrant is +, the cosine is –, and the tangent is - .

Again, as the revolving line moves from OB to OA', the perpendicular P N, diminishes until it becomes zero. Also, the base ON, increases in magnitude, until it finally coincides with OA', and .. = – OP, But the angle now described is 180°. Hence we have

OP 1
Sin 180° = = 0, cos 180o = - D = - 1,

0 tan 180° = Hence in the second quadrant The sine changes in magnitude from 1 to 0, and is positive. The cosine changes in magnitude from 0 to 1, and is negative. The tangent changes in magnitude from o to 0, and is negative.

And in the same way may we trace the changes of magnitude and sign in the third and fourth quadrants.

Thus we shall find

(3.) In the third quadrantThe sine changes in magnitude from 0 to 1, and is negative. The cosine changes in magnitude from 1 to 0, and is negative. The tangent changes in magnitude from 0 to 0, and is positive.

(4.) In the fourth quadrantThe sine changes in magnitude from 1 to 0, and is negative. The cosine changes in magnitude from 0 to 1, and is positive. The tangent changes in magnitude from co to 0, and is negative.

Moreover, as the cosecant, secant, and cotangent are respectively the reciprocals of the sine, cosine, and tangent, it follows that their signs will follow respectively the latter, and that their magnitudes will be their reciprocals.

CHAPTER IV. TRIGONOMETRICAL RATIOS CONTINUED. ARITHMETICAL VALUES

OF THE TRIGONOMETRICAL RATIOS OF 30°, 45°, 60°, &c. 12. To prove that sin A = cos (90° - A), and that

cos A = sin (90° - A).

Using the same figure as in Art. 5, we have

Sin a

PM
-AP = cos APM.

But LAPM = 90° - A,
.. Sin A = cos (90° – A),
and similarly,

cos A = sin (90° – A),
tan A = cot (90° – A),
cot A = tan (90° - A),
sec A = cosec (90° – A), A
cosec A = sec (90° - A).
13. Ratios of 45°.

In the last figure, suppose Z PAM = 45°, then also
L APM = 90° – 45o = 45°. And hence Z PAM = LAPM,
and ... PM = AM (Euc. I., 6).
Hence, also—
AP or NAM” + PM” = N2 AM or 12 PM”.

.. AP = AM 12 or PM 2.
Hence we have-

PM PM_
Sin 45o = sin PAM = AP = PM 15. = 75.

1 = cos 45o, by Art. 8.

PM PM Tan 45o = tan PAM = ū = D = 1 = cot 45°, by Art. 8.

AP AMZ
Sec 45o = sec PAM = M = AM = 12 = cosec 45°,

by Art. 8.
14. Ratios of 30° and 60°.

In the same diagram, suppose Z PAM = 30', then ZAPM = 90° - 30o = 60°.

Hence, if we conceive another triangle equal in every respect to APM to be described on the other side of AM, the whole would form an equilateral triangle whose side is AP.

Hence, PM = { AP.

Now AM = NAP? – PM', .. AM = VAP? – (4 AP):

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Art. 8.
15. To show that sin (180° - A) = sin A,
cos (180° - A) = – cos A,

Let 2 AOP, = A,

And let the revolving line describe an

angle AOP2 = 180° * A N,

N, A - A;

Then LA'OP, = 180° - (180° - A) = A;

Hence, 2 AOP1 = A 'OPZ.

Hence, also (Euc. I., 26), if P N, P,N, be drawn perpendicular to AA', PAN, = P2N2, ON, = - ON, We have therefore,

P_P,N, _ PN.
Sin (180° – A) = sin AOP, = 252 = 31 = sin A.
Cos (180° – A) = cos AOP, =

D ON, - - ON, = - cos A.

P2 = - OD = And similarly, Tan (180° - A) = - tan A, cot (180° - A) = – cot A. Sec (180° - A) = - sec A, cosec (180° - A) = cosec A.

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