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Angles which together make up two right-angles are said to be Supplementary to one another: and each angle is called the Supplement of the other.
Thus, the pairs of angles 30° and 150°, 47° 13' 29" and 132° 46' 31" are supplementary pairs. If A denote any angle, 180° – A denotes its supplement.
The trigonometrical in terms of the corresponding geometrical angle.
14. Let l'O1, K'OK be two lines at right-angles cutting in 0.
Let a line revolve from 01, as initial position, towards the perpendicular OK. Then, when the revolver
is between OI and OK, it is in the 1st Quadrant.
15. To find the general expression for the trigonometrical angle, when the final position of the revolving line is given.
Let OF be the final position of the revolving line corresponding to any angle.
1. Let OF, be in the 1st Quadrant. Let the acute angle IOF, be A.
Then, the revolving line may have described any number of complete revolutions, bringing it back to 01, before finally resting at OF.
Hence the trigonometrical angle 10 F, is n. 360° + A where n is any whole (positive) number.
2. Let OF, be in the 2nd Quadrant. Let the acute angle I'OF, be A.
Then, any number of complete revolutions, i.e. n. 360°, may be first described, bringing the revolver back to 01.
Now I0F, + I'OF,= 180o..I0F,= 180° - I'OF,= 180o - A.
n. 360° + 180° A = (2n + 1) 180° – A. 3. Let OF, be in the 3rd Quadrant. Let the acute angle I'OF, be A. Then, the revolving line may be supposed first, to make n complete revolutions from Ol back to OI;
then, to revolve from OI to OI', through 180°; and finally, to revolve from Ol' to 0F3, through angle A. Thus, the trigonometrical angle 10 F, is
n.360° + 180° + A = (2n + 1) 180° + A.
4. Let OF, be in the 4th Quadrant.
first, to make (n-1) complete revolutions from or back to OI;
then, to revolve from OI to OI', through 180°; and finally, to revolve from Ol' to OF, through 180° – A. Thus, the trigonometrical angle 10 F. is
(n − 1) 360° + 180° + 180° – A = n. 360° – A*.
16. By the last article, we see that we may express any trigonometrical angle in terms of the geometrical acute angle which the final position of the revolving line makes with the initial position (produced if necessary).
Summing up the results, when the final line is in 1st Quad., trig. angle = even multiple of 180° + an acute angle. 2nd Quad., trig. angle = odd multiple of 180° - an acute angle. 3rd Quad., trig. angle = odd multiple of 180° + an acute angle. 4th Quad., trig. angle = even multiple of 180° - an acute angle.
These results should be carefully noted and remembered t.
§ 2. AREAS.
17. In order to measure an area, it is necessary to express the ratio it bears to some known area ; just as to measure a length or a weight, it is necessary to express the ratio it bears to a known length or weight.
* In these four results, n is understood to be 0 or some positive integer. An explanation will be given later according to which n may be positive or negative.
+ The student should observe that to speak of an angle as being in a certain Quadrant is misleading: it is the revolving line which passes through the several Quadrants, and whose final position in one or other of the Quadrants indicates the magnitude of the angle.
18. To find the area of a rectangle, when the lengths of its adjacent sides are known. Let ABCD be a rectangle.
D Let AB contain m parts, each equal to AP; and let AD contain n parts, each equal to AQ.
From the points of division draw lines parallel to the sides of the rectangle.
B Then the whole rectangle ABCD is divided into small rectangles—the adjacent sides of each of these rectangles being equal to AP and AQ.
And the number of these rectangles is m x n.
rect. whose adj. sides are m times AP and n times AQ, = mn times rect. whose adj. sides are AP and AQ.
19. In the above proposition, suppose that AP = AQ. Then the rectangles, into which ABCD is divided, are squares.
Let AP or AQ be taken as the unit of length. Then, the measure of AB in terms of this unit is m, and the measure of AD in terms of this unit is n.
If we take the square on the unit of length as the unit of area, then the measure of ABCD in terms of this unit is mn.
Thus, the measure of the area of a rectangle is equal to the product of the measures of the lengths of its two adjacent sides.
20. The student should observe that an area is measured by the product of two numbers which measure lengths.
This is expressed by saying that area is of the second dimension in length.
21. The conclusion of Art. 19 may be expressed more briefly
The area of a rectangle=product of the lengths of its sides.
But arithmetically we can only multiply a quantity by an abstract number. If we extend the meaning of the term multiplication beyond that understood in Arithmetic, we may speak of physically multiplying one physical quantity by another, whenever the arithmetical multiplication of the measures of these quantities gives us the measure of some other physical quantity depending on these. It should be clearly understood that we are in this case extending the meaning of the word multiplication.
The area of a triangle. 22. By Euc. I. 41, if a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram is double of the triangle. P A Q P
Now, let ABC be any triangle,-acute-angled, obtuse-angled or right-angled.
From the vertex A, draw AL perpendicular to the base BC produced if necessary.
If the angle C is acute, I will fall between B and C, if
obtuse, L will fall beyond C, if
right, L will fall on C. Through A draw PAQ parallel to BC; and draw BP, CQ parallel to LA.
Then, in each case, the triangle A BC and the parallelogram PBCQ are on the same base BC, and between the same parallels BC, PQ,
.. the A ABC is į the parallelogram PBCQ. But the rectangle PBCQ is measured by the product of the measures of BC and BP; i.e. of BC and AL, .. the A ABC = } the product BC, AL
= } (base x altitude).